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Properties

Label 2-115-115.103-c1-0-1
Degree 22
Conductor 115115
Sign −0.549+0.835i-0.549 + 0.835i
Analytic cond. 0.9182790.918279
Root an. cond. 0.9582690.958269
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  + (−0.579 + 2.66i)2-s + (−2.04 + 1.52i)3-s + (−4.93 − 2.25i)4-s + (2.12 + 0.709i)5-s + (−2.89 − 6.32i)6-s + (−0.134 + 1.87i)7-s + (5.60 − 7.48i)8-s + (0.990 − 3.37i)9-s + (−3.11 + 5.23i)10-s + (−0.338 + 0.526i)11-s + (13.5 − 2.94i)12-s + (−2.62 + 0.187i)13-s + (−4.92 − 1.44i)14-s + (−5.41 + 1.79i)15-s + (9.57 + 11.0i)16-s + (0.245 − 0.657i)17-s + ⋯
L(s)  = 1  + (−0.409 + 1.88i)2-s + (−1.17 + 0.883i)3-s + (−2.46 − 1.12i)4-s + (0.948 + 0.317i)5-s + (−1.17 − 2.58i)6-s + (−0.0507 + 0.709i)7-s + (1.98 − 2.64i)8-s + (0.330 − 1.12i)9-s + (−0.985 + 1.65i)10-s + (−0.102 + 0.158i)11-s + (3.90 − 0.850i)12-s + (−0.727 + 0.0520i)13-s + (−1.31 − 0.386i)14-s + (−1.39 + 0.463i)15-s + (2.39 + 2.76i)16-s + (0.0595 − 0.159i)17-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=((−0.549+0.835i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+1/2) L(s)=((−0.549+0.835i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: −0.549+0.835i-0.549 + 0.835i
Analytic conductor: 0.9182790.918279
Root analytic conductor: 0.9582690.958269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ115(103,⋅)\chi_{115} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 115, ( :1/2), −0.549+0.835i)(2,\ 115,\ (\ :1/2),\ -0.549 + 0.835i)

Particular Values

L(1)L(1) ≈\approx 0.238440−0.441954i0.238440 - 0.441954i
L(12)L(\frac12) ≈\approx 0.238440−0.441954i0.238440 - 0.441954i
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)
bad5 1+(−2.12−0.709i)T 1 + (-2.12 - 0.709i)T
23 1+(3.33−3.44i)T 1 + (3.33 - 3.44i)T
good2 1+(0.579−2.66i)T+(−1.81−0.830i)T2 1 + (0.579 - 2.66i)T + (-1.81 - 0.830i)T^{2}
3 1+(2.04−1.52i)T+(0.845−2.87i)T2 1 + (2.04 - 1.52i)T + (0.845 - 2.87i)T^{2}
7 1+(0.134−1.87i)T+(−6.92−0.996i)T2 1 + (0.134 - 1.87i)T + (-6.92 - 0.996i)T^{2}
11 1+(0.338−0.526i)T+(−4.56−10.0i)T2 1 + (0.338 - 0.526i)T + (-4.56 - 10.0i)T^{2}
13 1+(2.62−0.187i)T+(12.8−1.85i)T2 1 + (2.62 - 0.187i)T + (12.8 - 1.85i)T^{2}
17 1+(−0.245+0.657i)T+(−12.8−11.1i)T2 1 + (-0.245 + 0.657i)T + (-12.8 - 11.1i)T^{2}
19 1+(0.374−0.819i)T+(−12.4−14.3i)T2 1 + (0.374 - 0.819i)T + (-12.4 - 14.3i)T^{2}
29 1+(0.285−0.130i)T+(18.9−21.9i)T2 1 + (0.285 - 0.130i)T + (18.9 - 21.9i)T^{2}
31 1+(−0.816−5.68i)T+(−29.7+8.73i)T2 1 + (-0.816 - 5.68i)T + (-29.7 + 8.73i)T^{2}
37 1+(−0.127−0.233i)T+(−20.0+31.1i)T2 1 + (-0.127 - 0.233i)T + (-20.0 + 31.1i)T^{2}
41 1+(−6.57+1.92i)T+(34.4−22.1i)T2 1 + (-6.57 + 1.92i)T + (34.4 - 22.1i)T^{2}
43 1+(−6.98−9.32i)T+(−12.1+41.2i)T2 1 + (-6.98 - 9.32i)T + (-12.1 + 41.2i)T^{2}
47 1+(−3.96+3.96i)T−47iT2 1 + (-3.96 + 3.96i)T - 47iT^{2}
53 1+(−6.60−0.472i)T+(52.4+7.54i)T2 1 + (-6.60 - 0.472i)T + (52.4 + 7.54i)T^{2}
59 1+(2.23+1.93i)T+(8.39+58.3i)T2 1 + (2.23 + 1.93i)T + (8.39 + 58.3i)T^{2}
61 1+(9.02−1.29i)T+(58.5−17.1i)T2 1 + (9.02 - 1.29i)T + (58.5 - 17.1i)T^{2}
67 1+(13.7+2.99i)T+(60.9+27.8i)T2 1 + (13.7 + 2.99i)T + (60.9 + 27.8i)T^{2}
71 1+(−5.95+3.82i)T+(29.4−64.5i)T2 1 + (-5.95 + 3.82i)T + (29.4 - 64.5i)T^{2}
73 1+(6.69−2.49i)T+(55.1−47.8i)T2 1 + (6.69 - 2.49i)T + (55.1 - 47.8i)T^{2}
79 1+(−6.21+7.16i)T+(−11.2−78.1i)T2 1 + (-6.21 + 7.16i)T + (-11.2 - 78.1i)T^{2}
83 1+(0.0389−0.0212i)T+(44.8−69.8i)T2 1 + (0.0389 - 0.0212i)T + (44.8 - 69.8i)T^{2}
89 1+(0.678−4.72i)T+(−85.3−25.0i)T2 1 + (0.678 - 4.72i)T + (-85.3 - 25.0i)T^{2}
97 1+(−9.14−4.99i)T+(52.4+81.6i)T2 1 + (-9.14 - 4.99i)T + (52.4 + 81.6i)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

−14.65745264394187198687032646583, −13.70154623428484176025425458207, −12.30244978617833379347798410960, −10.61010733143732331733347908285, −9.770442921644334363667495895585, −9.044652026886008630603296683745, −7.43449902408080320240271637166, −6.15373520705704231085976440578, −5.60118328484108183443265467650, −4.64052954795198168329691996913, 0.75913600923154005491167631284, 2.26968813677609194912601584614, 4.43377062670301981073560843590, 5.81082135733700591368015361680, 7.53622107096943701102983558551, 9.051574769954332686871080584979, 10.19235413413752190475205913328, 10.80167234644681439949626079854, 11.91374420698017730388777543247, 12.59200559494016912784518518104

Graph of the ZZ-function along the critical line