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Properties

Label 2-115-23.13-c3-0-20
Degree 22
Conductor 115115
Sign −0.117−0.993i-0.117 - 0.993i
Analytic cond. 6.785216.78521
Root an. cond. 2.604842.60484
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.78 − 3.20i)2-s + (0.620 + 0.398i)3-s + (−1.42 + 9.93i)4-s + (2.07 − 4.54i)5-s + (−0.446 − 3.10i)6-s + (−1.32 − 0.388i)7-s + (7.27 − 4.67i)8-s + (−10.9 − 24.0i)9-s + (−20.3 + 5.98i)10-s + (−14.8 + 17.1i)11-s + (−4.85 + 5.59i)12-s + (−74.9 + 22.0i)13-s + (2.43 + 5.32i)14-s + (3.10 − 1.99i)15-s + (41.7 + 12.2i)16-s + (11.2 + 77.9i)17-s + ⋯
L(s)  = 1  + (−0.983 − 1.13i)2-s + (0.119 + 0.0767i)3-s + (−0.178 + 1.24i)4-s + (0.185 − 0.406i)5-s + (−0.0303 − 0.211i)6-s + (−0.0713 − 0.0209i)7-s + (0.321 − 0.206i)8-s + (−0.407 − 0.891i)9-s + (−0.644 + 0.189i)10-s + (−0.407 + 0.470i)11-s + (−0.116 + 0.134i)12-s + (−1.60 + 0.469i)13-s + (0.0464 + 0.101i)14-s + (0.0534 − 0.0343i)15-s + (0.652 + 0.191i)16-s + (0.159 + 1.11i)17-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=((−0.117−0.993i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+3/2) L(s)=((−0.117−0.993i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: −0.117−0.993i-0.117 - 0.993i
Analytic conductor: 6.785216.78521
Root analytic conductor: 2.604842.60484
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ115(36,⋅)\chi_{115} (36, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 115, ( :3/2), −0.117−0.993i)(2,\ 115,\ (\ :3/2),\ -0.117 - 0.993i)

Particular Values

L(2)L(2) ≈\approx 3.99528×10−5+4.49470×10−5i3.99528\times10^{-5} + 4.49470\times10^{-5}i
L(12)L(\frac12) ≈\approx 3.99528×10−5+4.49470×10−5i3.99528\times10^{-5} + 4.49470\times10^{-5}i
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+(−2.07+4.54i)T 1 + (-2.07 + 4.54i)T
23 1+(34.9−104.i)T 1 + (34.9 - 104. i)T
good2 1+(2.78+3.20i)T+(−1.13+7.91i)T2 1 + (2.78 + 3.20i)T + (-1.13 + 7.91i)T^{2}
3 1+(−0.620−0.398i)T+(11.2+24.5i)T2 1 + (-0.620 - 0.398i)T + (11.2 + 24.5i)T^{2}
7 1+(1.32+0.388i)T+(288.+185.i)T2 1 + (1.32 + 0.388i)T + (288. + 185. i)T^{2}
11 1+(14.8−17.1i)T+(−189.−1.31e3i)T2 1 + (14.8 - 17.1i)T + (-189. - 1.31e3i)T^{2}
13 1+(74.9−22.0i)T+(1.84e3−1.18e3i)T2 1 + (74.9 - 22.0i)T + (1.84e3 - 1.18e3i)T^{2}
17 1+(−11.2−77.9i)T+(−4.71e3+1.38e3i)T2 1 + (-11.2 - 77.9i)T + (-4.71e3 + 1.38e3i)T^{2}
19 1+(−15.1+105.i)T+(−6.58e3−1.93e3i)T2 1 + (-15.1 + 105. i)T + (-6.58e3 - 1.93e3i)T^{2}
29 1+(−37.5−261.i)T+(−2.34e4+6.87e3i)T2 1 + (-37.5 - 261. i)T + (-2.34e4 + 6.87e3i)T^{2}
31 1+(169.−108.i)T+(1.23e4−2.70e4i)T2 1 + (169. - 108. i)T + (1.23e4 - 2.70e4i)T^{2}
37 1+(−55.8−122.i)T+(−3.31e4+3.82e4i)T2 1 + (-55.8 - 122. i)T + (-3.31e4 + 3.82e4i)T^{2}
41 1+(−154.+337.i)T+(−4.51e4−5.20e4i)T2 1 + (-154. + 337. i)T + (-4.51e4 - 5.20e4i)T^{2}
43 1+(376.+242.i)T+(3.30e4+7.23e4i)T2 1 + (376. + 242. i)T + (3.30e4 + 7.23e4i)T^{2}
47 1+376.T+1.03e5T2 1 + 376.T + 1.03e5T^{2}
53 1+(236.+69.5i)T+(1.25e5+8.04e4i)T2 1 + (236. + 69.5i)T + (1.25e5 + 8.04e4i)T^{2}
59 1+(93.6−27.5i)T+(1.72e5−1.11e5i)T2 1 + (93.6 - 27.5i)T + (1.72e5 - 1.11e5i)T^{2}
61 1+(−650.+417.i)T+(9.42e4−2.06e5i)T2 1 + (-650. + 417. i)T + (9.42e4 - 2.06e5i)T^{2}
67 1+(147.+169.i)T+(−4.28e4+2.97e5i)T2 1 + (147. + 169. i)T + (-4.28e4 + 2.97e5i)T^{2}
71 1+(−178.−205.i)T+(−5.09e4+3.54e5i)T2 1 + (-178. - 205. i)T + (-5.09e4 + 3.54e5i)T^{2}
73 1+(−56.5+393.i)T+(−3.73e5−1.09e5i)T2 1 + (-56.5 + 393. i)T + (-3.73e5 - 1.09e5i)T^{2}
79 1+(17.1−5.03i)T+(4.14e5−2.66e5i)T2 1 + (17.1 - 5.03i)T + (4.14e5 - 2.66e5i)T^{2}
83 1+(−362.−793.i)T+(−3.74e5+4.32e5i)T2 1 + (-362. - 793. i)T + (-3.74e5 + 4.32e5i)T^{2}
89 1+(535.+344.i)T+(2.92e5+6.41e5i)T2 1 + (535. + 344. i)T + (2.92e5 + 6.41e5i)T^{2}
97 1+(−241.+527.i)T+(−5.97e5−6.89e5i)T2 1 + (-241. + 527. i)T + (-5.97e5 - 6.89e5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−12.21091698475040698813473214711, −11.20226840065720622148381285895, −10.00706127796421050251651776530, −9.367531168822833278132917647139, −8.457681728309608362451593618917, −7.01346605814738011434599463480, −5.16544046597213070894008942469, −3.32203496942244072392678008414, −1.84792415876142234700126695734, −0.00004112996519286253218648726, 2.67621712119839734428916389165, 5.16398845258513166350917429852, 6.27325056959775251881272233185, 7.68337230807013327637699582075, 8.007069620910015031203990717996, 9.570564005272895045417092233151, 10.20083432807247326583734543391, 11.60498366018056294564443952958, 12.99382304853253867454427088533, 14.33138712292448877282841328185

Graph of the ZZ-function along the critical line