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Properties

Label 2-234-117.25-c1-0-5
Degree $2$
Conductor $234$
Sign $0.999 + 0.00672i$
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 + (−0.987 + 1.42i)3-s + (0.499 + 0.866i)4-s + (2.53 − 1.46i)5-s + (1.56 − 0.738i)6-s + (−1.90 − 1.10i)7-s − 0.999i·8-s + (−1.05 − 2.81i)9-s − 2.93·10-s + (4.47 + 2.58i)11-s + (−1.72 − 0.143i)12-s + (2.72 + 2.36i)13-s + (1.10 + 1.90i)14-s + (−0.420 + 5.06i)15-s + (−0.5 + 0.866i)16-s + 2.31·17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.569 + 0.821i)3-s + (0.249 + 0.433i)4-s + (1.13 − 0.655i)5-s + (0.639 − 0.301i)6-s + (−0.721 − 0.416i)7-s − 0.353i·8-s + (−0.350 − 0.936i)9-s − 0.927·10-s + (1.34 + 0.779i)11-s + (−0.498 − 0.0414i)12-s + (0.756 + 0.654i)13-s + (0.294 + 0.510i)14-s + (−0.108 + 1.30i)15-s + (−0.125 + 0.216i)16-s + 0.560·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.999 + 0.00672i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.999 + 0.00672i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.939801 - 0.00316114i\)
\(L(\frac12)\) \(\approx\) \(0.939801 - 0.00316114i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.987 - 1.42i)T \)
13 \( 1 + (-2.72 - 2.36i)T \)
good5 \( 1 + (-2.53 + 1.46i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.90 + 1.10i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.47 - 2.58i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 2.31T + 17T^{2} \)
19 \( 1 + 5.16iT - 19T^{2} \)
23 \( 1 + (-4.19 - 7.26i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.72 + 8.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.38 - 3.11i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.646iT - 37T^{2} \)
41 \( 1 + (0.674 - 0.389i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.74 - 3.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.79 + 2.76i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.68T + 53T^{2} \)
59 \( 1 + (-2.59 + 1.49i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.432 + 0.748i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.68 - 5.58i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.9iT - 71T^{2} \)
73 \( 1 - 4.27iT - 73T^{2} \)
79 \( 1 + (-1.52 + 2.63i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.19 + 1.84i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.34iT - 89T^{2} \)
97 \( 1 + (1.29 + 0.745i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.90359761565341945855652930097, −11.15748952145781708496375855521, −9.882854880427594147433227460446, −9.513585468672231774765538462400, −8.882541155078824731682631848701, −6.94242723384915894869528261546, −6.11686319727270000994111088148, −4.76006656287523172108565448603, −3.51112082262510937953662554516, −1.37294533765844972084907933263, 1.38589631668939481940868051337, 3.05141189846872143604909323886, 5.60068826747182151902636549193, 6.25749038393394653816838277435, 6.79632784614119182486712130100, 8.273460906700014648020247648246, 9.198824386212028918206691442785, 10.33686954685385643246276526107, 10.98276120954054199027423224860, 12.21969206416393879183587435073

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