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Properties

Label 2-234-117.41-c1-0-10
Degree $2$
Conductor $234$
Sign $0.878 + 0.476i$
Analytic cond. $1.86849$
Root an. cond. $1.36693$
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  + (−0.965 + 0.258i)2-s + (1.60 − 0.647i)3-s + (0.866 − 0.499i)4-s + (2.84 − 0.763i)5-s + (−1.38 + 1.04i)6-s + (−1.37 − 1.37i)7-s + (−0.707 + 0.707i)8-s + (2.16 − 2.07i)9-s + (−2.55 + 1.47i)10-s + (0.207 + 0.774i)11-s + (1.06 − 1.36i)12-s + (−3.58 − 0.369i)13-s + (1.68 + 0.975i)14-s + (4.08 − 3.06i)15-s + (0.500 − 0.866i)16-s + (−4.02 + 6.96i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.927 − 0.373i)3-s + (0.433 − 0.249i)4-s + (1.27 − 0.341i)5-s + (−0.565 + 0.424i)6-s + (−0.521 − 0.521i)7-s + (−0.249 + 0.249i)8-s + (0.720 − 0.693i)9-s + (−0.807 + 0.466i)10-s + (0.0625 + 0.233i)11-s + (0.308 − 0.393i)12-s + (−0.994 − 0.102i)13-s + (0.451 + 0.260i)14-s + (1.05 − 0.792i)15-s + (0.125 − 0.216i)16-s + (−0.975 + 1.68i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.476i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.878 + 0.476i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.878 + 0.476i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31821 - 0.334631i\)
\(L(\frac12)\) \(\approx\) \(1.31821 - 0.334631i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-1.60 + 0.647i)T \)
13 \( 1 + (3.58 + 0.369i)T \)
good5 \( 1 + (-2.84 + 0.763i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.37 + 1.37i)T + 7iT^{2} \)
11 \( 1 + (-0.207 - 0.774i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (4.02 - 6.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.18 + 4.41i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 6.99T + 23T^{2} \)
29 \( 1 + (-6.82 - 3.93i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.619 - 2.31i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.09 - 7.82i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (3.66 + 3.66i)T + 41iT^{2} \)
43 \( 1 - 2.33iT - 43T^{2} \)
47 \( 1 + (5.14 + 1.37i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 - 8.27iT - 53T^{2} \)
59 \( 1 + (8.52 + 2.28i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 4.93T + 61T^{2} \)
67 \( 1 + (-0.549 + 0.549i)T - 67iT^{2} \)
71 \( 1 + (2.57 - 0.689i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-5.68 - 5.68i)T + 73iT^{2} \)
79 \( 1 + (-0.554 - 0.960i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.614 + 2.29i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (2.84 + 0.761i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.59 + 2.59i)T - 97iT^{2} \)
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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

−12.42943212484892516788898008334, −10.67191780647538184973093738397, −9.951418219643264636821172269098, −9.117095840521486538157897653617, −8.426672943273344445927162380366, −6.99516096156211890117155259340, −6.46911669981743307906616803411, −4.77930826791968926589559932029, −2.86963670248523741993018189944, −1.58144345385466811831595302942, 2.20951009244079053766876995144, 2.96753116998392828038895073325, 4.90646862467217129018284809716, 6.36661002261329958814286064036, 7.36926338038589454773030106427, 8.708153648266285168840934254018, 9.496283946850151939372933625213, 9.893572324423413702362766650158, 10.96076097048235364003894943500, 12.26949238350492626481044711186

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