Properties

Label 2-114-19.16-c3-0-4
Degree $2$
Conductor $114$
Sign $0.945 + 0.325i$
Analytic cond. $6.72621$
Root an. cond. $2.59349$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.53 − 1.28i)2-s + (2.81 − 1.02i)3-s + (0.694 + 3.93i)4-s + (0.810 − 4.59i)5-s + (−5.63 − 2.05i)6-s + (15.9 + 27.6i)7-s + (4.00 − 6.92i)8-s + (6.89 − 5.78i)9-s + (−7.15 + 5.99i)10-s + (4.72 − 8.17i)11-s + (6 + 10.3i)12-s + (24.0 + 8.75i)13-s + (11.0 − 62.8i)14-s + (−2.43 − 13.7i)15-s + (−15.0 + 5.47i)16-s + (70.5 + 59.1i)17-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (0.542 − 0.197i)3-s + (0.0868 + 0.492i)4-s + (0.0724 − 0.411i)5-s + (−0.383 − 0.139i)6-s + (0.861 + 1.49i)7-s + (0.176 − 0.306i)8-s + (0.255 − 0.214i)9-s + (−0.226 + 0.189i)10-s + (0.129 − 0.224i)11-s + (0.144 + 0.249i)12-s + (0.512 + 0.186i)13-s + (0.211 − 1.19i)14-s + (−0.0418 − 0.237i)15-s + (−0.234 + 0.0855i)16-s + (1.00 + 0.844i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $0.945 + 0.325i$
Analytic conductor: \(6.72621\)
Root analytic conductor: \(2.59349\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :3/2),\ 0.945 + 0.325i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.63244 - 0.273400i\)
\(L(\frac12)\) \(\approx\) \(1.63244 - 0.273400i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.53 + 1.28i)T \)
3 \( 1 + (-2.81 + 1.02i)T \)
19 \( 1 + (14.5 + 81.5i)T \)
good5 \( 1 + (-0.810 + 4.59i)T + (-117. - 42.7i)T^{2} \)
7 \( 1 + (-15.9 - 27.6i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-4.72 + 8.17i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-24.0 - 8.75i)T + (1.68e3 + 1.41e3i)T^{2} \)
17 \( 1 + (-70.5 - 59.1i)T + (853. + 4.83e3i)T^{2} \)
23 \( 1 + (16.1 + 91.6i)T + (-1.14e4 + 4.16e3i)T^{2} \)
29 \( 1 + (-223. + 187. i)T + (4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (-33.4 - 57.8i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 292.T + 5.06e4T^{2} \)
41 \( 1 + (217. - 79.1i)T + (5.27e4 - 4.43e4i)T^{2} \)
43 \( 1 + (88.0 - 499. i)T + (-7.47e4 - 2.71e4i)T^{2} \)
47 \( 1 + (218. - 183. i)T + (1.80e4 - 1.02e5i)T^{2} \)
53 \( 1 + (8.22 + 46.6i)T + (-1.39e5 + 5.09e4i)T^{2} \)
59 \( 1 + (358. + 300. i)T + (3.56e4 + 2.02e5i)T^{2} \)
61 \( 1 + (103. + 589. i)T + (-2.13e5 + 7.76e4i)T^{2} \)
67 \( 1 + (-48.4 + 40.6i)T + (5.22e4 - 2.96e5i)T^{2} \)
71 \( 1 + (122. - 696. i)T + (-3.36e5 - 1.22e5i)T^{2} \)
73 \( 1 + (-340. + 123. i)T + (2.98e5 - 2.50e5i)T^{2} \)
79 \( 1 + (77.0 - 28.0i)T + (3.77e5 - 3.16e5i)T^{2} \)
83 \( 1 + (-75.3 - 130. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (382. + 139. i)T + (5.40e5 + 4.53e5i)T^{2} \)
97 \( 1 + (-35.6 - 29.8i)T + (1.58e5 + 8.98e5i)T^{2} \)
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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

−12.76519090859554827410593262727, −12.05014720428231096967339618651, −11.03939846418898619061472144160, −9.658733157719451955879829725792, −8.495384944710770822779957289621, −8.285644021780928542548413482459, −6.37440715628879402973206011595, −4.81485736173430295021070647777, −2.90534669292161937375465310031, −1.50030417340714553026549643549, 1.32259351341030443868145710390, 3.56559135414331972728139742250, 5.07283429163255764167205269734, 6.87830619780860193743389171864, 7.68047038357047288046547357441, 8.676844076889718723725027460401, 10.20041146208971007955502805212, 10.54039619775722057965050881796, 11.97607657493042517154461518506, 13.76378368182265799377083756490

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