Properties

Label 2-115-115.9-c1-0-2
Degree $2$
Conductor $115$
Sign $-0.260 - 0.965i$
Analytic cond. $0.918279$
Root an. cond. $0.958269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.320 + 1.09i)2-s + (−1.24 + 1.07i)3-s + (0.591 − 0.379i)4-s + (0.206 + 2.22i)5-s + (−1.57 − 1.01i)6-s + (−1.40 − 0.643i)7-s + (2.32 + 2.01i)8-s + (−0.0408 + 0.283i)9-s + (−2.36 + 0.940i)10-s + (−4.00 − 1.17i)11-s + (−0.326 + 1.11i)12-s + (5.39 − 2.46i)13-s + (0.251 − 1.74i)14-s + (−2.65 − 2.54i)15-s + (−0.872 + 1.91i)16-s + (−0.798 + 1.24i)17-s + ⋯
L(s)  = 1  + (0.226 + 0.772i)2-s + (−0.718 + 0.622i)3-s + (0.295 − 0.189i)4-s + (0.0923 + 0.995i)5-s + (−0.644 − 0.414i)6-s + (−0.532 − 0.243i)7-s + (0.822 + 0.712i)8-s + (−0.0136 + 0.0946i)9-s + (−0.748 + 0.297i)10-s + (−1.20 − 0.354i)11-s + (−0.0941 + 0.320i)12-s + (1.49 − 0.682i)13-s + (0.0671 − 0.466i)14-s + (−0.686 − 0.658i)15-s + (−0.218 + 0.477i)16-s + (−0.193 + 0.301i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.260 - 0.965i$
Analytic conductor: \(0.918279\)
Root analytic conductor: \(0.958269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1/2),\ -0.260 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.634101 + 0.827788i\)
\(L(\frac12)\) \(\approx\) \(0.634101 + 0.827788i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.206 - 2.22i)T \)
23 \( 1 + (-3.89 + 2.79i)T \)
good2 \( 1 + (-0.320 - 1.09i)T + (-1.68 + 1.08i)T^{2} \)
3 \( 1 + (1.24 - 1.07i)T + (0.426 - 2.96i)T^{2} \)
7 \( 1 + (1.40 + 0.643i)T + (4.58 + 5.29i)T^{2} \)
11 \( 1 + (4.00 + 1.17i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (-5.39 + 2.46i)T + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (0.798 - 1.24i)T + (-7.06 - 15.4i)T^{2} \)
19 \( 1 + (-6.07 + 3.90i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (-2.24 - 1.44i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (-0.638 + 0.736i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (6.34 + 0.912i)T + (35.5 + 10.4i)T^{2} \)
41 \( 1 + (0.434 + 3.01i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (7.10 - 6.15i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 - 3.03iT - 47T^{2} \)
53 \( 1 + (1.91 + 0.872i)T + (34.7 + 40.0i)T^{2} \)
59 \( 1 + (1.42 + 3.13i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (1.61 - 1.86i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (1.00 + 3.43i)T + (-56.3 + 36.2i)T^{2} \)
71 \( 1 + (6.31 - 1.85i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-4.77 - 7.43i)T + (-30.3 + 66.4i)T^{2} \)
79 \( 1 + (-2.53 - 5.54i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (4.51 + 0.648i)T + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (-5.73 - 6.62i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-13.8 + 1.98i)T + (93.0 - 27.3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.89507222928628587925192926356, −13.25262726018232314538785341170, −11.27184949721236008849737021732, −10.81985501713508225099621021311, −10.09403888119501575344827787584, −8.169321672402893301368581231038, −6.97397456019995078930003693891, −5.97356174049360330593512406561, −5.09073308329074596681505024098, −3.08768587483890860855792661000, 1.43601252075190412096410591844, 3.44770526249277777013793138487, 5.20732277473543094492765342548, 6.44487943126177764940010797427, 7.70112587391338419328616280564, 9.144653911159615365617624416602, 10.38027365600734553203187783888, 11.63812928441579143517907323449, 12.09232516168566133776739075427, 13.10800234380933171148848700618

Graph of the $Z$-function along the critical line