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Properties

Label 2-114-57.50-c1-0-6
Degree $2$
Conductor $114$
Sign $0.102 + 0.994i$
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 + (−1.72 − 0.158i)6-s − 0.449·7-s + 0.999·8-s + (−1.00 − 2.82i)9-s + (−1.22 − 0.707i)10-s + 3.14i·11-s + (0.724 + 1.57i)12-s + (−3 − 1.73i)13-s + (0.224 + 0.389i)14-s + (0.224 − 2.43i)15-s + (−0.5 − 0.866i)16-s + (0.550 − 0.317i)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 + (−0.704 − 0.0648i)6-s − 0.169·7-s + 0.353·8-s + (−0.333 − 0.942i)9-s + (−0.387 − 0.223i)10-s + 0.948i·11-s + (0.209 + 0.454i)12-s + (−0.832 − 0.480i)13-s + (0.0600 + 0.104i)14-s + (0.0580 − 0.629i)15-s + (−0.125 − 0.216i)16-s + (0.133 − 0.0770i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $0.102 + 0.994i$
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: \((2,\ 114,\ (\ :1/2),\ 0.102 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.782780 - 0.706501i\)
\(L(\frac12)\) \(\approx\) \(0.782780 - 0.706501i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.17 - 2.98i)T \)
good5 \( 1 + (-1.22 + 0.707i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.449T + 7T^{2} \)
11 \( 1 - 3.14iT - 11T^{2} \)
13 \( 1 + (3 + 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.550 + 0.317i)T + (8.5 - 14.7i)T^{2} \)
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 \( 1 + 7.70iT - 37T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.44 - 7.70i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (11.5 + 6.68i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.44 + 9.43i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.72 + 9.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.775 + 1.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.17 - 1.25i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.39 + 7.61i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.34 - 4.24i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 17.0iT - 83T^{2} \)
89 \( 1 + (3.55 - 6.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.84 + 1.64i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.97315686066031126629809881678, −12.60446884044509840228902903061, −11.42488568715242049699646505126, −9.840858066176326772679822925622, −9.312890924620304496582503963563, −7.928640561208158473309675077217, −7.00914468353950443903852884434, −5.24990632257554510788593133287, −3.21534186766200558312398054343, −1.68465876703108235790528508369, 2.82772192678536030945170781749, 4.65012206407404933622648897726, 5.95679877052569384345613007563, 7.34105964881883823157273450015, 8.633204051433608548029331777161, 9.487398685572345572993080132627, 10.34980780601817069103649344349, 11.45583897787186427116524262228, 13.27346793441228356201047879966, 14.07355260441014396025208814727

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