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Properties

Label 2-220-55.17-c1-0-5
Degree 22
Conductor 220220
Sign 0.0835+0.996i0.0835 + 0.996i
Analytic cond. 1.756701.75670
Root an. cond. 1.325401.32540
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.464 + 0.0735i)3-s + (−0.794 − 2.09i)5-s + (0.413 − 2.61i)7-s + (−2.64 + 0.858i)9-s + (1.70 − 2.84i)11-s + (0.780 − 1.53i)13-s + (0.522 + 0.911i)15-s + (−1.07 − 2.11i)17-s + (2.28 + 1.65i)19-s + 1.24i·21-s + (4.09 − 4.09i)23-s + (−3.73 + 3.31i)25-s + (2.41 − 1.23i)27-s + (−1.79 + 1.30i)29-s + (1.63 + 5.02i)31-s + ⋯
L(s)  = 1  + (−0.267 + 0.0424i)3-s + (−0.355 − 0.934i)5-s + (0.156 − 0.987i)7-s + (−0.881 + 0.286i)9-s + (0.514 − 0.857i)11-s + (0.216 − 0.425i)13-s + (0.134 + 0.235i)15-s + (−0.261 − 0.512i)17-s + (0.523 + 0.380i)19-s + 0.271i·21-s + (0.854 − 0.854i)23-s + (−0.747 + 0.663i)25-s + (0.465 − 0.237i)27-s + (−0.333 + 0.242i)29-s + (0.293 + 0.902i)31-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((0.0835+0.996i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0835 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1/2) L(s)=((0.0835+0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0835 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: 0.0835+0.996i0.0835 + 0.996i
Analytic conductor: 1.756701.75670
Root analytic conductor: 1.325401.32540
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ220(17,⋅)\chi_{220} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1/2), 0.0835+0.996i)(2,\ 220,\ (\ :1/2),\ 0.0835 + 0.996i)

Particular Values

L(1)L(1) ≈\approx 0.703051−0.646564i0.703051 - 0.646564i
L(12)L(\frac12) ≈\approx 0.703051−0.646564i0.703051 - 0.646564i
L(32)L(\frac{3}{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)
bad2 1 1
5 1+(0.794+2.09i)T 1 + (0.794 + 2.09i)T
11 1+(−1.70+2.84i)T 1 + (-1.70 + 2.84i)T
good3 1+(0.464−0.0735i)T+(2.85−0.927i)T2 1 + (0.464 - 0.0735i)T + (2.85 - 0.927i)T^{2}
7 1+(−0.413+2.61i)T+(−6.65−2.16i)T2 1 + (-0.413 + 2.61i)T + (-6.65 - 2.16i)T^{2}
13 1+(−0.780+1.53i)T+(−7.64−10.5i)T2 1 + (-0.780 + 1.53i)T + (-7.64 - 10.5i)T^{2}
17 1+(1.07+2.11i)T+(−9.99+13.7i)T2 1 + (1.07 + 2.11i)T + (-9.99 + 13.7i)T^{2}
19 1+(−2.28−1.65i)T+(5.87+18.0i)T2 1 + (-2.28 - 1.65i)T + (5.87 + 18.0i)T^{2}
23 1+(−4.09+4.09i)T−23iT2 1 + (-4.09 + 4.09i)T - 23iT^{2}
29 1+(1.79−1.30i)T+(8.96−27.5i)T2 1 + (1.79 - 1.30i)T + (8.96 - 27.5i)T^{2}
31 1+(−1.63−5.02i)T+(−25.0+18.2i)T2 1 + (-1.63 - 5.02i)T + (-25.0 + 18.2i)T^{2}
37 1+(2.56+0.405i)T+(35.1+11.4i)T2 1 + (2.56 + 0.405i)T + (35.1 + 11.4i)T^{2}
41 1+(6.37−8.77i)T+(−12.6−38.9i)T2 1 + (6.37 - 8.77i)T + (-12.6 - 38.9i)T^{2}
43 1+(−4.89−4.89i)T+43iT2 1 + (-4.89 - 4.89i)T + 43iT^{2}
47 1+(1.64+10.3i)T+(−44.6+14.5i)T2 1 + (1.64 + 10.3i)T + (-44.6 + 14.5i)T^{2}
53 1+(−8.77−4.47i)T+(31.1+42.8i)T2 1 + (-8.77 - 4.47i)T + (31.1 + 42.8i)T^{2}
59 1+(2.11+2.91i)T+(−18.2+56.1i)T2 1 + (2.11 + 2.91i)T + (-18.2 + 56.1i)T^{2}
61 1+(−11.5−3.74i)T+(49.3+35.8i)T2 1 + (-11.5 - 3.74i)T + (49.3 + 35.8i)T^{2}
67 1+(−4.30−4.30i)T+67iT2 1 + (-4.30 - 4.30i)T + 67iT^{2}
71 1+(−0.522+1.60i)T+(−57.4−41.7i)T2 1 + (-0.522 + 1.60i)T + (-57.4 - 41.7i)T^{2}
73 1+(14.3+2.26i)T+(69.4+22.5i)T2 1 + (14.3 + 2.26i)T + (69.4 + 22.5i)T^{2}
79 1+(−2.50−7.70i)T+(−63.9+46.4i)T2 1 + (-2.50 - 7.70i)T + (-63.9 + 46.4i)T^{2}
83 1+(−8.02+4.08i)T+(48.7−67.1i)T2 1 + (-8.02 + 4.08i)T + (48.7 - 67.1i)T^{2}
89 1+11.6iT−89T2 1 + 11.6iT - 89T^{2}
97 1+(−4.57+8.98i)T+(−57.0−78.4i)T2 1 + (-4.57 + 8.98i)T + (-57.0 - 78.4i)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

−11.88030139421175968245246995455, −11.22726323914039347087210890372, −10.26697157388226112337651021005, −8.878224108543834864043430315927, −8.256784215194765479596499415808, −7.02579269026141870343419094597, −5.66343233663628813772758264111, −4.66130832598507651984754492008, −3.33225061927011388659344795072, −0.870204952360649524822876012924, 2.34687084713542113902114177822, 3.74332503405092731026501853295, 5.35524377818991351005875694975, 6.39850607150078339014216549156, 7.35730665364908957588931754895, 8.671483962914801097433431428057, 9.510950576869478702914700234533, 10.82098477091492432930647260677, 11.66258090136669461476022735014, 12.11577928268090652344505823893

Graph of the ZZ-function along the critical line