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Properties

Label 2-115-115.114-c4-0-15
Degree $2$
Conductor $115$
Sign $-0.872 - 0.489i$
Analytic cond. $11.8875$
Root an. cond. $3.44783$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.17i·2-s − 1.68i·3-s − 10.7·4-s + (−10.7 + 22.5i)5-s + 8.72·6-s + 91.0·7-s + 26.9i·8-s + 78.1·9-s + (−116. − 55.7i)10-s + 122. i·11-s + 18.1i·12-s − 261. i·13-s + 471. i·14-s + (38.0 + 18.1i)15-s − 312.·16-s − 361.·17-s + ⋯
L(s)  = 1  + 1.29i·2-s − 0.187i·3-s − 0.674·4-s + (−0.430 + 0.902i)5-s + 0.242·6-s + 1.85·7-s + 0.421i·8-s + 0.964·9-s + (−1.16 − 0.557i)10-s + 1.01i·11-s + 0.126i·12-s − 1.54i·13-s + 2.40i·14-s + (0.169 + 0.0807i)15-s − 1.21·16-s − 1.25·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.489i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.872 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.872 - 0.489i$
Analytic conductor: \(11.8875\)
Root analytic conductor: \(3.44783\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :2),\ -0.872 - 0.489i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.505944 + 1.93605i\)
\(L(\frac12)\) \(\approx\) \(0.505944 + 1.93605i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (10.7 - 22.5i)T \)
23 \( 1 + (34.8 - 527. i)T \)
good2 \( 1 - 5.17iT - 16T^{2} \)
3 \( 1 + 1.68iT - 81T^{2} \)
7 \( 1 - 91.0T + 2.40e3T^{2} \)
11 \( 1 - 122. iT - 1.46e4T^{2} \)
13 \( 1 + 261. iT - 2.85e4T^{2} \)
17 \( 1 + 361.T + 8.35e4T^{2} \)
19 \( 1 - 477. iT - 1.30e5T^{2} \)
29 \( 1 - 985.T + 7.07e5T^{2} \)
31 \( 1 + 675.T + 9.23e5T^{2} \)
37 \( 1 + 230.T + 1.87e6T^{2} \)
41 \( 1 + 1.08e3T + 2.82e6T^{2} \)
43 \( 1 - 2.03e3T + 3.41e6T^{2} \)
47 \( 1 + 2.21e3iT - 4.87e6T^{2} \)
53 \( 1 - 745.T + 7.89e6T^{2} \)
59 \( 1 - 703.T + 1.21e7T^{2} \)
61 \( 1 + 570. iT - 1.38e7T^{2} \)
67 \( 1 - 3.64e3T + 2.01e7T^{2} \)
71 \( 1 + 759.T + 2.54e7T^{2} \)
73 \( 1 + 879. iT - 2.83e7T^{2} \)
79 \( 1 - 2.23e3iT - 3.89e7T^{2} \)
83 \( 1 + 6.08e3T + 4.74e7T^{2} \)
89 \( 1 + 25.1iT - 6.27e7T^{2} \)
97 \( 1 - 3.90e3T + 8.85e7T^{2} \)
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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

−13.71805045671232074913203701260, −12.26448454959736993396165109429, −11.14146371932991866710278703436, −10.22804501188276941226043784661, −8.303064045805328696745860474094, −7.66490326771781680403555931683, −6.96104568633496713323925995509, −5.47339191184111550358861639071, −4.32010027486929037666503373345, −1.95678391717539233397637606784, 0.952633448465848130392419319017, 2.08093323978243459159484124303, 4.27972231482974966965672433579, 4.66007370833379695372545306486, 6.96766908657424250266543717358, 8.525170094290322457756345358508, 9.188066869551593832281757144895, 10.82633078692789077200067836771, 11.30166642576068607443426694313, 12.09797923469213446168024790246

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