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Properties

Label 2-114-57.56-c1-0-1
Degree $2$
Conductor $114$
Sign $0.596 - 0.802i$
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 + (−1.5 + 0.866i)3-s + 4-s + 3.46i·5-s + (−1.5 + 0.866i)6-s + 7-s + 8-s + (1.5 − 2.59i)9-s + 3.46i·10-s − 3.46i·11-s + (−1.5 + 0.866i)12-s − 1.73i·13-s + 14-s + (−2.99 − 5.19i)15-s + 16-s + 1.73i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.866 + 0.499i)3-s + 0.5·4-s + 1.54i·5-s + (−0.612 + 0.353i)6-s + 0.377·7-s + 0.353·8-s + (0.5 − 0.866i)9-s + 1.09i·10-s − 1.04i·11-s + (−0.433 + 0.249i)12-s − 0.480i·13-s + 0.267·14-s + (−0.774 − 1.34i)15-s + 0.250·16-s + 0.420i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $0.596 - 0.802i$
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: \((2,\ 114,\ (\ :1/2),\ 0.596 - 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09994 + 0.553375i\)
\(L(\frac12)\) \(\approx\) \(1.09994 + 0.553375i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 0.866i)T \)
19 \( 1 + (4 - 1.73i)T \)
good5 \( 1 - 3.46iT - 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 - 1.73iT - 17T^{2} \)
23 \( 1 + 5.19iT - 23T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + 10.3iT - 31T^{2} \)
37 \( 1 - 6.92iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + 3T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 8.66iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 - 6.92iT - 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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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

−13.93865406482178534730604431979, −12.61561510748546313828220353693, −11.43843241121292595298890662752, −10.82738199385273632155144207714, −10.11430042684651296706286539675, −8.112262278083392832402569165284, −6.54564424096198188129094551060, −5.99522944989182225075727631750, −4.36832954540764876987581180628, −2.99063705687272663952860070260, 1.64369780833473190775040709017, 4.58512735759355047775599264373, 5.07148813490284239032473666909, 6.53910247390834776155477036816, 7.76232100624168519610049007081, 9.118119754959359622108277338484, 10.58246001556314574441209511667, 11.86405326257156272124322486984, 12.38098109882178152909384304628, 13.15489901751515059955542535017

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