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Properties

Label 2-234-117.103-c1-0-3
Degree $2$
Conductor $234$
Sign $0.706 - 0.707i$
Analytic cond. $1.86849$
Root an. cond. $1.36693$
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  + (0.866 − 0.5i)2-s + (−0.579 + 1.63i)3-s + (0.499 − 0.866i)4-s + (3.32 + 1.91i)5-s + (0.314 + 1.70i)6-s + (−2.91 + 1.68i)7-s − 0.999i·8-s + (−2.32 − 1.89i)9-s + 3.83·10-s + (−1.72 + 0.995i)11-s + (1.12 + 1.31i)12-s + (3.49 − 0.880i)13-s + (−1.68 + 2.91i)14-s + (−5.05 + 4.31i)15-s + (−0.5 − 0.866i)16-s + 2.60·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.334 + 0.942i)3-s + (0.249 − 0.433i)4-s + (1.48 + 0.857i)5-s + (0.128 + 0.695i)6-s + (−1.10 + 0.636i)7-s − 0.353i·8-s + (−0.776 − 0.630i)9-s + 1.21·10-s + (−0.520 + 0.300i)11-s + (0.324 + 0.380i)12-s + (0.969 − 0.244i)13-s + (−0.450 + 0.779i)14-s + (−1.30 + 1.11i)15-s + (−0.125 − 0.216i)16-s + 0.630·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.706 - 0.707i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.706 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57429 + 0.652928i\)
\(L(\frac12)\) \(\approx\) \(1.57429 + 0.652928i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.579 - 1.63i)T \)
13 \( 1 + (-3.49 + 0.880i)T \)
good5 \( 1 + (-3.32 - 1.91i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.91 - 1.68i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.72 - 0.995i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 2.60T + 17T^{2} \)
19 \( 1 + 1.99iT - 19T^{2} \)
23 \( 1 + (-2.13 + 3.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.37 + 7.57i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.57 - 2.64i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.08iT - 37T^{2} \)
41 \( 1 + (7.13 + 4.12i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.13 + 8.88i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.22 + 2.44i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.16T + 53T^{2} \)
59 \( 1 + (6.33 + 3.65i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.63 - 9.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.90 - 2.83i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.94iT - 71T^{2} \)
73 \( 1 + 8.41iT - 73T^{2} \)
79 \( 1 + (-2.97 - 5.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.03 - 1.17i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 6.28iT - 89T^{2} \)
97 \( 1 + (8.92 - 5.15i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.32965220508879958296985903473, −11.16603330349314167157988469679, −10.20371519982264707872484203633, −9.904903600551229124411524118908, −8.833792789509025457682747610380, −6.63160030017111797997589659194, −5.99774277640459988646154583221, −5.17684796044451166490612489766, −3.43855660486355485308663168415, −2.53667735561754364405172692476, 1.46678260321413163402632281744, 3.22899845136571553868141625409, 5.13969813407119300144359663660, 5.97464459292821714101500710895, 6.62770271906413540285217847288, 7.902048028967581353837033433382, 9.077153564038840853151108430242, 10.12295325612518342005268557569, 11.26872888490478276080157782076, 12.67743216913891803115338029828

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