Properties

Label 2-114-57.8-c1-0-3
Degree $2$
Conductor $114$
Sign $0.974 + 0.223i$
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 − 1.41i)3-s + (−0.499 − 0.866i)4-s + (−1.22 − 0.707i)5-s + (0.724 + 1.57i)6-s + 4.44·7-s + 0.999·8-s + (−1.00 − 2.82i)9-s + (1.22 − 0.707i)10-s − 0.317i·11-s + (−1.72 − 0.158i)12-s + (−3 + 1.73i)13-s + (−2.22 + 3.85i)14-s + (−2.22 + 1.02i)15-s + (−0.5 + 0.866i)16-s + (5.44 + 3.14i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.577 − 0.816i)3-s + (−0.249 − 0.433i)4-s + (−0.547 − 0.316i)5-s + (0.295 + 0.642i)6-s + 1.68·7-s + 0.353·8-s + (−0.333 − 0.942i)9-s + (0.387 − 0.223i)10-s − 0.0958i·11-s + (−0.497 − 0.0458i)12-s + (−0.832 + 0.480i)13-s + (−0.594 + 1.02i)14-s + (−0.574 + 0.264i)15-s + (−0.125 + 0.216i)16-s + (1.32 + 0.763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $0.974 + 0.223i$
Analytic conductor: \(0.910294\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :1/2),\ 0.974 + 0.223i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02955 - 0.116559i\)
\(L(\frac12)\) \(\approx\) \(1.02955 - 0.116559i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.41i)T \)
19 \( 1 + (4.17 - 1.25i)T \)
good5 \( 1 + (1.22 + 0.707i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 4.44T + 7T^{2} \)
11 \( 1 + 0.317iT - 11T^{2} \)
13 \( 1 + (3 - 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-5.44 - 3.14i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (6.12 - 3.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.22 - 2.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 + 0.778iT - 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.449 - 0.778i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.57 + 3.21i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.550 - 0.953i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.27 - 5.67i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.22 - 5.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.17 - 2.98i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.39 + 9.35i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.34 - 4.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.1iT - 83T^{2} \)
89 \( 1 + (8.44 + 14.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.8 + 6.84i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.99446802528484295516182565359, −12.39828240888905651922789484436, −11.75622884263661958555354534012, −10.27261375222713417678822363460, −8.726384060449015888927563690374, −8.042270286817653117544590641933, −7.36690078948811546224439219263, −5.75397282070503442631068894885, −4.23139950921729108461214450473, −1.71825552365282122115149657755, 2.41441984541700376014803708653, 4.05455537546136089266178135595, 5.12257764103634430479328663228, 7.75173649679101607519044460897, 8.112689184043381657184008671386, 9.531196663781359117394746118593, 10.52184275178788379248776729973, 11.35644484555350922335922556438, 12.25506221700387901940439388035, 13.89983379193558356891455946713

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