Properties

Label 2-114-57.56-c1-0-5
Degree $2$
Conductor $114$
Sign $-0.662 + 0.749i$
Analytic cond. $0.910294$
Root an. cond. $0.954093$
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  − 2-s + (−1 − 1.41i)3-s + 4-s − 1.41i·5-s + (1 + 1.41i)6-s − 4·7-s − 8-s + (−1.00 + 2.82i)9-s + 1.41i·10-s − 5.65i·11-s + (−1 − 1.41i)12-s − 4.24i·13-s + 4·14-s + (−2.00 + 1.41i)15-s + 16-s + 2.82i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.577 − 0.816i)3-s + 0.5·4-s − 0.632i·5-s + (0.408 + 0.577i)6-s − 1.51·7-s − 0.353·8-s + (−0.333 + 0.942i)9-s + 0.447i·10-s − 1.70i·11-s + (−0.288 − 0.408i)12-s − 1.17i·13-s + 1.06·14-s + (−0.516 + 0.365i)15-s + 0.250·16-s + 0.685i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $-0.662 + 0.749i$
Analytic conductor: \(0.910294\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :1/2),\ -0.662 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.187891 - 0.416840i\)
\(L(\frac12)\) \(\approx\) \(0.187891 - 0.416840i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1 + 1.41i)T \)
19 \( 1 + (-1 - 4.24i)T \)
good5 \( 1 + 1.41iT - 5T^{2} \)
7 \( 1 + 4T + 7T^{2} \)
11 \( 1 + 5.65iT - 11T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
23 \( 1 + 1.41iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 4.24iT - 31T^{2} \)
37 \( 1 + 4.24iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + 1.41iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 4.24iT - 79T^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 8.48iT - 97T^{2} \)
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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

−12.86554969930770695368190381367, −12.39727882625190788016740595340, −11.01841174469738338022426905919, −10.13004918113045919660344999085, −8.732450706857503879431768376690, −7.88739778826632931859306112493, −6.36644239338775995376605019153, −5.72842745695832435114139827872, −3.13805947770897882212007565114, −0.66816671282063161139195356884, 2.97323078393205506761813742798, 4.67072515898436690904446070940, 6.56900575356071618745082935569, 6.99016355562247833280236354883, 9.206044813673965073257274287356, 9.687257480288723321972873330012, 10.56265684308273267318571520182, 11.72666334402644106659205396212, 12.59996735577287581826995145004, 14.17461496914506874884294786783

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