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Properties

Label 2-115-115.107-c1-0-1
Degree 22
Conductor 115115
Sign −0.999+0.0320i-0.999 + 0.0320i
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  + (−1.23 + 0.927i)2-s + (−1.46 − 0.547i)3-s + (0.110 − 0.377i)4-s + (1.69 + 1.46i)5-s + (2.32 − 0.683i)6-s + (−4.02 + 0.876i)7-s + (−0.868 − 2.32i)8-s + (−0.409 − 0.354i)9-s + (−3.45 − 0.238i)10-s + (−4.51 + 0.649i)11-s + (−0.369 + 0.493i)12-s + (−0.844 + 3.88i)13-s + (4.17 − 4.82i)14-s + (−1.68 − 3.07i)15-s + (3.89 + 2.50i)16-s + (0.141 − 0.258i)17-s + ⋯
L(s)  = 1  + (−0.875 + 0.655i)2-s + (−0.848 − 0.316i)3-s + (0.0553 − 0.188i)4-s + (0.757 + 0.653i)5-s + (0.950 − 0.278i)6-s + (−1.52 + 0.331i)7-s + (−0.307 − 0.823i)8-s + (−0.136 − 0.118i)9-s + (−1.09 − 0.0754i)10-s + (−1.36 + 0.195i)11-s + (−0.106 + 0.142i)12-s + (−0.234 + 1.07i)13-s + (1.11 − 1.28i)14-s + (−0.435 − 0.793i)15-s + (0.974 + 0.626i)16-s + (0.0342 − 0.0627i)17-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=((−0.999+0.0320i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0320i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+1/2) L(s)=((−0.999+0.0320i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0320i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: −0.999+0.0320i-0.999 + 0.0320i
Analytic conductor: 0.9182790.918279
Root analytic conductor: 0.9582690.958269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ115(107,⋅)\chi_{115} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 115, ( :1/2), −0.999+0.0320i)(2,\ 115,\ (\ :1/2),\ -0.999 + 0.0320i)

Particular Values

L(1)L(1) ≈\approx 0.00352219−0.219418i0.00352219 - 0.219418i
L(12)L(\frac12) ≈\approx 0.00352219−0.219418i0.00352219 - 0.219418i
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+(−1.69−1.46i)T 1 + (-1.69 - 1.46i)T
23 1+(0.292+4.78i)T 1 + (0.292 + 4.78i)T
good2 1+(1.23−0.927i)T+(0.563−1.91i)T2 1 + (1.23 - 0.927i)T + (0.563 - 1.91i)T^{2}
3 1+(1.46+0.547i)T+(2.26+1.96i)T2 1 + (1.46 + 0.547i)T + (2.26 + 1.96i)T^{2}
7 1+(4.02−0.876i)T+(6.36−2.90i)T2 1 + (4.02 - 0.876i)T + (6.36 - 2.90i)T^{2}
11 1+(4.51−0.649i)T+(10.5−3.09i)T2 1 + (4.51 - 0.649i)T + (10.5 - 3.09i)T^{2}
13 1+(0.844−3.88i)T+(−11.8−5.40i)T2 1 + (0.844 - 3.88i)T + (-11.8 - 5.40i)T^{2}
17 1+(−0.141+0.258i)T+(−9.19−14.3i)T2 1 + (-0.141 + 0.258i)T + (-9.19 - 14.3i)T^{2}
19 1+(−3.49−1.02i)T+(15.9+10.2i)T2 1 + (-3.49 - 1.02i)T + (15.9 + 10.2i)T^{2}
29 1+(−0.403−1.37i)T+(−24.3+15.6i)T2 1 + (-0.403 - 1.37i)T + (-24.3 + 15.6i)T^{2}
31 1+(2.77−6.08i)T+(−20.3−23.4i)T2 1 + (2.77 - 6.08i)T + (-20.3 - 23.4i)T^{2}
37 1+(−0.220−3.08i)T+(−36.6+5.26i)T2 1 + (-0.220 - 3.08i)T + (-36.6 + 5.26i)T^{2}
41 1+(6.08+7.02i)T+(−5.83+40.5i)T2 1 + (6.08 + 7.02i)T + (-5.83 + 40.5i)T^{2}
43 1+(−0.482+1.29i)T+(−32.4−28.1i)T2 1 + (-0.482 + 1.29i)T + (-32.4 - 28.1i)T^{2}
47 1+(−9.04−9.04i)T+47iT2 1 + (-9.04 - 9.04i)T + 47iT^{2}
53 1+(0.184+0.847i)T+(−48.2+22.0i)T2 1 + (0.184 + 0.847i)T + (-48.2 + 22.0i)T^{2}
59 1+(2.33+3.63i)T+(−24.5+53.6i)T2 1 + (2.33 + 3.63i)T + (-24.5 + 53.6i)T^{2}
61 1+(3.31+1.51i)T+(39.9+46.1i)T2 1 + (3.31 + 1.51i)T + (39.9 + 46.1i)T^{2}
67 1+(−0.270−0.361i)T+(−18.8+64.2i)T2 1 + (-0.270 - 0.361i)T + (-18.8 + 64.2i)T^{2}
71 1+(0.537−3.73i)T+(−68.1−20.0i)T2 1 + (0.537 - 3.73i)T + (-68.1 - 20.0i)T^{2}
73 1+(3.83−2.09i)T+(39.4−61.4i)T2 1 + (3.83 - 2.09i)T + (39.4 - 61.4i)T^{2}
79 1+(8.40−5.39i)T+(32.8−71.8i)T2 1 + (8.40 - 5.39i)T + (32.8 - 71.8i)T^{2}
83 1+(−0.939+0.0672i)T+(82.1−11.8i)T2 1 + (-0.939 + 0.0672i)T + (82.1 - 11.8i)T^{2}
89 1+(−1.99−4.35i)T+(−58.2+67.2i)T2 1 + (-1.99 - 4.35i)T + (-58.2 + 67.2i)T^{2}
97 1+(7.77+0.556i)T+(96.0+13.8i)T2 1 + (7.77 + 0.556i)T + (96.0 + 13.8i)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

−14.07739990784167071247466837154, −12.86326168598345903728913510960, −12.20560057379735038487167603726, −10.62466343367965654938539891039, −9.776512555257760426313406916505, −8.888166347715044326940320093396, −7.17812479139612480088898793836, −6.56974626535451590138818664523, −5.60240408996451414205564154346, −3.01932520409192900068422646256, 0.32863689174953200078330714399, 2.80574410052741156463498113576, 5.30010180892064442940163565049, 5.86612921384523157521295244970, 7.83390659027218753300128879357, 9.235081685196203040822623058868, 10.10859938293795059572800894404, 10.50384148311578455168583570828, 11.78526021828480528767783606242, 12.98739022728605615932502664518

Graph of the ZZ-function along the critical line