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Properties

Label 2-114-57.56-c1-0-7
Degree $2$
Conductor $114$
Sign $0.927 + 0.374i$
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  + 2-s + (1 − 1.41i)3-s + 4-s + 1.41i·5-s + (1 − 1.41i)6-s − 4·7-s + 8-s + (−1.00 − 2.82i)9-s + 1.41i·10-s + 5.65i·11-s + (1 − 1.41i)12-s − 4.24i·13-s − 4·14-s + (2.00 + 1.41i)15-s + 16-s − 2.82i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.577 − 0.816i)3-s + 0.5·4-s + 0.632i·5-s + (0.408 − 0.577i)6-s − 1.51·7-s + 0.353·8-s + (−0.333 − 0.942i)9-s + 0.447i·10-s + 1.70i·11-s + (0.288 − 0.408i)12-s − 1.17i·13-s − 1.06·14-s + (0.516 + 0.365i)15-s + 0.250·16-s − 0.685i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $0.927 + 0.374i$
Analytic conductor: \(0.910294\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :1/2),\ 0.927 + 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54269 - 0.299894i\)
\(L(\frac12)\) \(\approx\) \(1.54269 - 0.299894i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1 + 1.41i)T \)
19 \( 1 + (-1 - 4.24i)T \)
good5 \( 1 - 1.41iT - 5T^{2} \)
7 \( 1 + 4T + 7T^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
23 \( 1 - 1.41iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 4.24iT - 31T^{2} \)
37 \( 1 + 4.24iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 - 1.41iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 4.24iT - 79T^{2} \)
83 \( 1 + 2.82iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 8.48iT - 97T^{2} \)
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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

−13.25092394538342122552329217174, −12.78992616265785395876288403106, −11.93527617100321057580621015372, −10.28203756050024740552854529559, −9.433711982995668883810367783194, −7.56995308509356553363623823211, −6.94909488668822592649725569771, −5.77270130908789855328915338266, −3.64773973712878028607102946935, −2.52754969442324543840005105817, 3.00489611670191032575990616124, 4.04605938820785517167919607874, 5.51443916036930876837155240244, 6.73832795362808146064718524367, 8.587510014585810766821465324872, 9.275813778740862873684277615043, 10.54303899595967610980929629665, 11.60928323620627003007230293400, 13.02674724024610970578432370356, 13.53467691744317623608872832408

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