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Properties

Label 2-87-29.23-c1-0-3
Degree 22
Conductor 8787
Sign 0.994+0.103i0.994 + 0.103i
Analytic cond. 0.6946980.694698
Root an. cond. 0.8334850.833485
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  + (−0.0518 + 0.0650i)2-s + (0.222 − 0.974i)3-s + (0.443 + 1.94i)4-s + (1.20 − 1.50i)5-s + (0.0518 + 0.0650i)6-s + (−0.0765 + 0.335i)7-s + (−0.299 − 0.144i)8-s + (−0.900 − 0.433i)9-s + (0.0357 + 0.156i)10-s + (2.18 − 1.05i)11-s + 1.99·12-s + (−3.27 + 1.57i)13-s + (−0.0178 − 0.0223i)14-s + (−1.20 − 1.50i)15-s + (−3.56 + 1.71i)16-s − 6.15·17-s + ⋯
L(s)  = 1  + (−0.0366 + 0.0459i)2-s + (0.128 − 0.562i)3-s + (0.221 + 0.971i)4-s + (0.538 − 0.674i)5-s + (0.0211 + 0.0265i)6-s + (−0.0289 + 0.126i)7-s + (−0.105 − 0.0509i)8-s + (−0.300 − 0.144i)9-s + (0.0112 + 0.0494i)10-s + (0.658 − 0.317i)11-s + 0.575·12-s + (−0.907 + 0.436i)13-s + (−0.00476 − 0.00597i)14-s + (−0.310 − 0.389i)15-s + (−0.891 + 0.429i)16-s − 1.49·17-s + ⋯

Functional equation

Λ(s)=(87s/2 ΓC(s) L(s)=((0.994+0.103i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(87s/2 ΓC(s+1/2) L(s)=((0.994+0.103i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8787    =    3⋅293 \cdot 29
Sign: 0.994+0.103i0.994 + 0.103i
Analytic conductor: 0.6946980.694698
Root analytic conductor: 0.8334850.833485
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ87(52,⋅)\chi_{87} (52, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 87, ( :1/2), 0.994+0.103i)(2,\ 87,\ (\ :1/2),\ 0.994 + 0.103i)

Particular Values

L(1)L(1) ≈\approx 1.05541−0.0545208i1.05541 - 0.0545208i
L(12)L(\frac12) ≈\approx 1.05541−0.0545208i1.05541 - 0.0545208i
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)
bad3 1+(−0.222+0.974i)T 1 + (-0.222 + 0.974i)T
29 1+(4.26−3.29i)T 1 + (4.26 - 3.29i)T
good2 1+(0.0518−0.0650i)T+(−0.445−1.94i)T2 1 + (0.0518 - 0.0650i)T + (-0.445 - 1.94i)T^{2}
5 1+(−1.20+1.50i)T+(−1.11−4.87i)T2 1 + (-1.20 + 1.50i)T + (-1.11 - 4.87i)T^{2}
7 1+(0.0765−0.335i)T+(−6.30−3.03i)T2 1 + (0.0765 - 0.335i)T + (-6.30 - 3.03i)T^{2}
11 1+(−2.18+1.05i)T+(6.85−8.60i)T2 1 + (-2.18 + 1.05i)T + (6.85 - 8.60i)T^{2}
13 1+(3.27−1.57i)T+(8.10−10.1i)T2 1 + (3.27 - 1.57i)T + (8.10 - 10.1i)T^{2}
17 1+6.15T+17T2 1 + 6.15T + 17T^{2}
19 1+(−0.300−1.31i)T+(−17.1+8.24i)T2 1 + (-0.300 - 1.31i)T + (-17.1 + 8.24i)T^{2}
23 1+(2.16+2.70i)T+(−5.11+22.4i)T2 1 + (2.16 + 2.70i)T + (-5.11 + 22.4i)T^{2}
31 1+(−6.85+8.59i)T+(−6.89−30.2i)T2 1 + (-6.85 + 8.59i)T + (-6.89 - 30.2i)T^{2}
37 1+(−2.78−1.33i)T+(23.0+28.9i)T2 1 + (-2.78 - 1.33i)T + (23.0 + 28.9i)T^{2}
41 1−5.28T+41T2 1 - 5.28T + 41T^{2}
43 1+(0.00526+0.00659i)T+(−9.56+41.9i)T2 1 + (0.00526 + 0.00659i)T + (-9.56 + 41.9i)T^{2}
47 1+(−5.22+2.51i)T+(29.3−36.7i)T2 1 + (-5.22 + 2.51i)T + (29.3 - 36.7i)T^{2}
53 1+(2.58−3.23i)T+(−11.7−51.6i)T2 1 + (2.58 - 3.23i)T + (-11.7 - 51.6i)T^{2}
59 1−5.76T+59T2 1 - 5.76T + 59T^{2}
61 1+(−1.38+6.08i)T+(−54.9−26.4i)T2 1 + (-1.38 + 6.08i)T + (-54.9 - 26.4i)T^{2}
67 1+(9.92+4.77i)T+(41.7+52.3i)T2 1 + (9.92 + 4.77i)T + (41.7 + 52.3i)T^{2}
71 1+(11.5−5.57i)T+(44.2−55.5i)T2 1 + (11.5 - 5.57i)T + (44.2 - 55.5i)T^{2}
73 1+(−0.286−0.359i)T+(−16.2+71.1i)T2 1 + (-0.286 - 0.359i)T + (-16.2 + 71.1i)T^{2}
79 1+(−11.2−5.41i)T+(49.2+61.7i)T2 1 + (-11.2 - 5.41i)T + (49.2 + 61.7i)T^{2}
83 1+(0.640+2.80i)T+(−74.7+36.0i)T2 1 + (0.640 + 2.80i)T + (-74.7 + 36.0i)T^{2}
89 1+(7.62−9.55i)T+(−19.8−86.7i)T2 1 + (7.62 - 9.55i)T + (-19.8 - 86.7i)T^{2}
97 1+(−0.606−2.65i)T+(−87.3+42.0i)T2 1 + (-0.606 - 2.65i)T + (-87.3 + 42.0i)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

−13.88625020839372426783470706555, −13.04612654294961150131848483169, −12.20951197224154527463370358444, −11.28268786247941478890160201829, −9.408569874164116857878163937355, −8.605981583085119117281117588126, −7.35118704892474149938195512250, −6.18236823293577027874684331002, −4.31830425762404274827130986333, −2.32574764180977286338758692199, 2.42845099258208896765894669285, 4.56719057115337065583188079935, 6.00361210190932933877974384582, 7.09332890571311192474067950348, 9.041094055793296856686353436235, 9.975440516733443496364393678535, 10.69629314685900320849090979998, 11.80263783270728393119184326482, 13.49621310297348168381822310243, 14.39736827457890762328597777300

Graph of the ZZ-function along the critical line