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Properties

Label 2-234-117.103-c1-0-9
Degree $2$
Conductor $234$
Sign $0.973 + 0.227i$
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 + (1.66 + 0.480i)3-s + (0.499 − 0.866i)4-s + (0.515 + 0.297i)5-s + (1.68 − 0.416i)6-s + (−1.45 + 0.838i)7-s − 0.999i·8-s + (2.53 + 1.59i)9-s + 0.594·10-s + (−0.416 + 0.240i)11-s + (1.24 − 1.20i)12-s + (−2.27 − 2.79i)13-s + (−0.838 + 1.45i)14-s + (0.714 + 0.742i)15-s + (−0.5 − 0.866i)16-s − 2.09·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.960 + 0.277i)3-s + (0.249 − 0.433i)4-s + (0.230 + 0.132i)5-s + (0.686 − 0.169i)6-s + (−0.548 + 0.316i)7-s − 0.353i·8-s + (0.846 + 0.532i)9-s + 0.188·10-s + (−0.125 + 0.0724i)11-s + (0.360 − 0.346i)12-s + (−0.630 − 0.776i)13-s + (−0.224 + 0.388i)14-s + (0.184 + 0.191i)15-s + (−0.125 − 0.216i)16-s − 0.507·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.227i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.973 + 0.227i$
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: \((2,\ 234,\ (\ :1/2),\ 0.973 + 0.227i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13855 - 0.246526i\)
\(L(\frac12)\) \(\approx\) \(2.13855 - 0.246526i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.66 - 0.480i)T \)
13 \( 1 + (2.27 + 2.79i)T \)
good5 \( 1 + (-0.515 - 0.297i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.45 - 0.838i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.416 - 0.240i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 2.09T + 17T^{2} \)
19 \( 1 + 0.480iT - 19T^{2} \)
23 \( 1 + (1.83 - 3.17i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.23 - 2.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.993 + 0.573i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.65iT - 37T^{2} \)
41 \( 1 + (-8.58 - 4.95i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.45 + 5.98i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.40 - 3.12i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.08T + 53T^{2} \)
59 \( 1 + (-8.13 - 4.69i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.90 + 6.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.4 - 7.19i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.51iT - 71T^{2} \)
73 \( 1 + 5.91iT - 73T^{2} \)
79 \( 1 + (-1.02 - 1.78i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.57 - 5.53i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.48iT - 89T^{2} \)
97 \( 1 + (-8.41 + 4.85i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.48235566612562226359361744002, −11.14959010777295727765195358777, −10.06733077209014353526528237174, −9.501276667028101183915521040867, −8.261227116370834689915837650831, −7.12834929686288526865171161808, −5.82102592374527322777891613318, −4.54936170535268934751209010602, −3.27195241860241752466794980364, −2.26076031267130636164631320936, 2.20041243820958489385173564026, 3.56344703688578581479097108876, 4.69444008951545229445954931308, 6.29441893844356062568527921009, 7.12032331964608550877654080367, 8.138948933392736250176010346745, 9.228600904119461894514039439408, 10.05540701826271799474639862294, 11.50070555154639086049904901647, 12.60127086902683813831774516150

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