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Properties

Label 2-115-115.102-c3-0-0
Degree 22
Conductor 115115
Sign 0.995−0.0900i0.995 - 0.0900i
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  + (0.371 − 5.19i)2-s + (−8.92 + 1.94i)3-s + (−18.8 − 2.71i)4-s + (−8.89 − 6.77i)5-s + (6.76 + 47.0i)6-s + (8.66 − 4.72i)7-s + (−12.2 + 56.3i)8-s + (51.2 − 23.4i)9-s + (−38.4 + 43.6i)10-s + (−37.0 − 32.0i)11-s + (173. − 12.4i)12-s + (−1.59 + 2.92i)13-s + (−21.3 − 46.7i)14-s + (92.4 + 43.2i)15-s + (141. + 41.5i)16-s + (53.9 + 40.3i)17-s + ⋯
L(s)  = 1  + (0.131 − 1.83i)2-s + (−1.71 + 0.373i)3-s + (−2.36 − 0.339i)4-s + (−0.795 − 0.606i)5-s + (0.460 + 3.20i)6-s + (0.467 − 0.255i)7-s + (−0.542 + 2.49i)8-s + (1.89 − 0.867i)9-s + (−1.21 + 1.37i)10-s + (−1.01 − 0.879i)11-s + (4.18 − 0.299i)12-s + (−0.0341 + 0.0624i)13-s + (−0.407 − 0.891i)14-s + (1.59 + 0.744i)15-s + (2.21 + 0.649i)16-s + (0.769 + 0.576i)17-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=((0.995−0.0900i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0900i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+3/2) L(s)=((0.995−0.0900i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0900i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: 0.995−0.0900i0.995 - 0.0900i
Analytic conductor: 6.785216.78521
Root analytic conductor: 2.604842.60484
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ115(102,⋅)\chi_{115} (102, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 115, ( :3/2), 0.995−0.0900i)(2,\ 115,\ (\ :3/2),\ 0.995 - 0.0900i)

Particular Values

L(2)L(2) ≈\approx 0.0767753+0.00346499i0.0767753 + 0.00346499i
L(12)L(\frac12) ≈\approx 0.0767753+0.00346499i0.0767753 + 0.00346499i
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+(8.89+6.77i)T 1 + (8.89 + 6.77i)T
23 1+(−2.23−110.i)T 1 + (-2.23 - 110. i)T
good2 1+(−0.371+5.19i)T+(−7.91−1.13i)T2 1 + (-0.371 + 5.19i)T + (-7.91 - 1.13i)T^{2}
3 1+(8.92−1.94i)T+(24.5−11.2i)T2 1 + (8.92 - 1.94i)T + (24.5 - 11.2i)T^{2}
7 1+(−8.66+4.72i)T+(185.−288.i)T2 1 + (-8.66 + 4.72i)T + (185. - 288. i)T^{2}
11 1+(37.0+32.0i)T+(189.+1.31e3i)T2 1 + (37.0 + 32.0i)T + (189. + 1.31e3i)T^{2}
13 1+(1.59−2.92i)T+(−1.18e3−1.84e3i)T2 1 + (1.59 - 2.92i)T + (-1.18e3 - 1.84e3i)T^{2}
17 1+(−53.9−40.3i)T+(1.38e3+4.71e3i)T2 1 + (-53.9 - 40.3i)T + (1.38e3 + 4.71e3i)T^{2}
19 1+(−9.07+63.0i)T+(−6.58e3−1.93e3i)T2 1 + (-9.07 + 63.0i)T + (-6.58e3 - 1.93e3i)T^{2}
29 1+(284.−40.8i)T+(2.34e4−6.87e3i)T2 1 + (284. - 40.8i)T + (2.34e4 - 6.87e3i)T^{2}
31 1+(−58.5+37.6i)T+(1.23e4−2.70e4i)T2 1 + (-58.5 + 37.6i)T + (1.23e4 - 2.70e4i)T^{2}
37 1+(−347.−129.i)T+(3.82e4+3.31e4i)T2 1 + (-347. - 129. i)T + (3.82e4 + 3.31e4i)T^{2}
41 1+(−20.3+44.6i)T+(−4.51e4−5.20e4i)T2 1 + (-20.3 + 44.6i)T + (-4.51e4 - 5.20e4i)T^{2}
43 1+(−11.1−51.3i)T+(−7.23e4+3.30e4i)T2 1 + (-11.1 - 51.3i)T + (-7.23e4 + 3.30e4i)T^{2}
47 1+(446.+446.i)T+1.03e5iT2 1 + (446. + 446. i)T + 1.03e5iT^{2}
53 1+(−74.9−137.i)T+(−8.04e4+1.25e5i)T2 1 + (-74.9 - 137. i)T + (-8.04e4 + 1.25e5i)T^{2}
59 1+(30.6+104.i)T+(−1.72e5+1.11e5i)T2 1 + (30.6 + 104. i)T + (-1.72e5 + 1.11e5i)T^{2}
61 1+(44.3+69.0i)T+(−9.42e4+2.06e5i)T2 1 + (44.3 + 69.0i)T + (-9.42e4 + 2.06e5i)T^{2}
67 1+(686.+49.1i)T+(2.97e5+4.28e4i)T2 1 + (686. + 49.1i)T + (2.97e5 + 4.28e4i)T^{2}
71 1+(−71.9−82.9i)T+(−5.09e4+3.54e5i)T2 1 + (-71.9 - 82.9i)T + (-5.09e4 + 3.54e5i)T^{2}
73 1+(−279.−374.i)T+(−1.09e5+3.73e5i)T2 1 + (-279. - 374. i)T + (-1.09e5 + 3.73e5i)T^{2}
79 1+(−820.+240.i)T+(4.14e5−2.66e5i)T2 1 + (-820. + 240. i)T + (4.14e5 - 2.66e5i)T^{2}
83 1+(179.−481.i)T+(−4.32e5−3.74e5i)T2 1 + (179. - 481. i)T + (-4.32e5 - 3.74e5i)T^{2}
89 1+(253.+162.i)T+(2.92e5+6.41e5i)T2 1 + (253. + 162. i)T + (2.92e5 + 6.41e5i)T^{2}
97 1+(−4.94−13.2i)T+(−6.89e5+5.97e5i)T2 1 + (-4.94 - 13.2i)T + (-6.89e5 + 5.97e5i)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

−12.72575229800917169773630597543, −11.59278874998007028598050782715, −11.31725081777797063586905645964, −10.51803762666355282630757820965, −9.426703696777318339565140854000, −7.893351833228260624955694143002, −5.55286609181535238832214609720, −4.78220741865826595567676677956, −3.62571898375426108553230376693, −1.09764304802572881672034702852, 0.06222806141869095087770374003, 4.47471416553086015497968449077, 5.36990970876392466849438751842, 6.33311895981247645948835508450, 7.43729027661344475710436040911, 7.88103825938489425126006476290, 9.887625521806449319900016968079, 11.10660597393510494178297095220, 12.24409284219888858364558624879, 13.03551673789915775624932616580

Graph of the ZZ-function along the critical line