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Properties

Label 2-114-57.8-c1-0-4
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 $1$

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 − 1.73i·6-s − 2·7-s + 0.999·8-s + (1.5 − 2.59i)9-s + (3 − 1.73i)10-s + 1.73i·11-s + (1.49 + 0.866i)12-s + (−3 + 1.73i)13-s + (1 − 1.73i)14-s + 6·15-s + (−0.5 + 0.866i)16-s + (−6 − 3.46i)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 − 0.707i·6-s − 0.755·7-s + 0.353·8-s + (0.5 − 0.866i)9-s + (0.948 − 0.547i)10-s + 0.522i·11-s + (0.433 + 0.250i)12-s + (−0.832 + 0.480i)13-s + (0.267 − 0.462i)14-s + 1.54·15-s + (−0.125 + 0.216i)16-s + (−1.45 − 0.840i)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: $\chi_{114} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 114,\ (\ :1/2),\ -0.740 + 0.671i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 4.33i)T \)
good5 \( 1 + (3 + 1.73i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 - 1.73iT - 11T^{2} \)
13 \( 1 + (3 - 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (6 + 3.46i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.92iT - 31T^{2} \)
37 \( 1 + 6.92iT - 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3 + 1.73i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.5 + 2.59i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.5 - 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (12 + 6.92i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.19iT - 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.5 - 4.33i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.90827632006148094617876878745, −12.10867606907445232844022519677, −11.18412967965461406980137132853, −9.817584436886253214074820769885, −8.998245159580685822868552706014, −7.56110063126240219339484692946, −6.57667224547453187125174561201, −4.99762765579652302841456530831, −4.09377601857051945711254367337, 0, 2.92474174787155679446208324715, 4.50093324347433417577487718006, 6.46395096170721997174046984607, 7.34965980248173268667283238865, 8.522355992659609080579137339979, 10.20701896490707379654839101008, 11.01410673724088045363400974489, 11.76088000692072397196604889310, 12.64180788202229429145452644977

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