Properties

Label 2-114-57.50-c1-0-0
Degree $2$
Conductor $114$
Sign $-0.740 - 0.671i$
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  + (0.5 + 0.866i)2-s + (−1.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−3 + 1.73i)5-s + (−1.5 − 0.866i)6-s + 7-s − 0.999·8-s + (1.5 − 2.59i)9-s + (−3 − 1.73i)10-s + 3.46i·11-s − 1.73i·12-s + (4.5 + 2.59i)13-s + (0.5 + 0.866i)14-s + (3 − 5.19i)15-s + (−0.5 − 0.866i)16-s + (3 − 1.73i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.866 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−1.34 + 0.774i)5-s + (−0.612 − 0.353i)6-s + 0.377·7-s − 0.353·8-s + (0.5 − 0.866i)9-s + (−0.948 − 0.547i)10-s + 1.04i·11-s − 0.499i·12-s + (1.24 + 0.720i)13-s + (0.133 + 0.231i)14-s + (0.774 − 1.34i)15-s + (−0.125 − 0.216i)16-s + (0.727 − 0.420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 - 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $-0.740 - 0.671i$
Analytic conductor: \(0.910294\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :1/2),\ -0.740 - 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.273905 + 0.709924i\)
\(L(\frac12)\) \(\approx\) \(0.273905 + 0.709924i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (1.5 - 0.866i)T \)
19 \( 1 + (4 - 1.73i)T \)
good5 \( 1 + (3 - 1.73i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 + (-4.5 - 2.59i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3 + 1.73i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + 5.19iT - 37T^{2} \)
41 \( 1 + (-6 - 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6 + 3.46i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.5 - 4.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.5 + 2.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6 + 3.46i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.42523747140850661344326232074, −12.78951134301813824973529789799, −11.73925569930681308939593260603, −11.18408077480697387861441470150, −9.925093853600671066882103237684, −8.320897757636082964727315934810, −7.18684934220337001163323860335, −6.23502792569374287118793849084, −4.61031991713176461845988323493, −3.75486857896117614295779222679, 0.925932476813854479131803856221, 3.67396725005845432295325936298, 4.96684836277158435161736630940, 6.16234758361217626460643157504, 7.893468515568327552261057284843, 8.643800337145997528836872995992, 10.72444268687988517463598989558, 11.16667191009346429714088634197, 12.20904673297579849637342308621, 12.82243495678826302919942471214

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