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Properties

Label 2-232-29.6-c1-0-0
Degree $2$
Conductor $232$
Sign $-0.517 - 0.855i$
Analytic cond. $1.85252$
Root an. cond. $1.36107$
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  + (0.0690 + 0.0157i)3-s + (−2.25 + 2.82i)5-s + (−0.686 + 3.00i)7-s + (−2.69 − 1.29i)9-s + (−1.30 − 2.70i)11-s + (−2.00 + 0.967i)13-s + (−0.199 + 0.159i)15-s + 7.63i·17-s + (7.68 − 1.75i)19-s + (−0.0947 + 0.196i)21-s + (4.03 + 5.06i)23-s + (−1.78 − 7.82i)25-s + (−0.332 − 0.264i)27-s + (−2.39 − 4.82i)29-s + (2.99 + 2.38i)31-s + ⋯
L(s)  = 1  + (0.0398 + 0.00910i)3-s + (−1.00 + 1.26i)5-s + (−0.259 + 1.13i)7-s + (−0.899 − 0.433i)9-s + (−0.392 − 0.814i)11-s + (−0.557 + 0.268i)13-s + (−0.0516 + 0.0411i)15-s + 1.85i·17-s + (1.76 − 0.402i)19-s + (−0.0206 + 0.0429i)21-s + (0.841 + 1.05i)23-s + (−0.357 − 1.56i)25-s + (−0.0638 − 0.0509i)27-s + (−0.444 − 0.895i)29-s + (0.538 + 0.429i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.517 - 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $-0.517 - 0.855i$
Analytic conductor: \(1.85252\)
Root analytic conductor: \(1.36107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 232,\ (\ :1/2),\ -0.517 - 0.855i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.366433 + 0.649942i\)
\(L(\frac12)\) \(\approx\) \(0.366433 + 0.649942i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (2.39 + 4.82i)T \)
good3 \( 1 + (-0.0690 - 0.0157i)T + (2.70 + 1.30i)T^{2} \)
5 \( 1 + (2.25 - 2.82i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (0.686 - 3.00i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (1.30 + 2.70i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (2.00 - 0.967i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 - 7.63iT - 17T^{2} \)
19 \( 1 + (-7.68 + 1.75i)T + (17.1 - 8.24i)T^{2} \)
23 \( 1 + (-4.03 - 5.06i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (-2.99 - 2.38i)T + (6.89 + 30.2i)T^{2} \)
37 \( 1 + (1.25 - 2.61i)T + (-23.0 - 28.9i)T^{2} \)
41 \( 1 - 4.52iT - 41T^{2} \)
43 \( 1 + (-3.17 + 2.53i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (2.52 + 5.23i)T + (-29.3 + 36.7i)T^{2} \)
53 \( 1 + (1.07 - 1.35i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + 14.4T + 59T^{2} \)
61 \( 1 + (-10.6 - 2.43i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (4.70 + 2.26i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (-1.43 + 0.691i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-1.08 + 0.867i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + (1.80 - 3.74i)T + (-49.2 - 61.7i)T^{2} \)
83 \( 1 + (-2.21 - 9.71i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-3.24 - 2.59i)T + (19.8 + 86.7i)T^{2} \)
97 \( 1 + (0.732 - 0.167i)T + (87.3 - 42.0i)T^{2} \)
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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

−12.15113662613181536567824010923, −11.57809195796759276094353581573, −10.86166673672723493769323208332, −9.574730543244189979058327102498, −8.501274734177248823810620936663, −7.61226420223455486941259023289, −6.39555744431569113847670348809, −5.48168481970944144837525388950, −3.51575989026128175217214981477, −2.82489955404563606095884519227, 0.60495225700205703780303212331, 3.09262881635611162265345274038, 4.61255330173643299516913937704, 5.21037891656873194278384243886, 7.31497870089332730691909732418, 7.65863628341551187146042938503, 8.930890950919825201015677742909, 9.830040431158185414848426029127, 11.06421644732243054920267999971, 11.94168835886939753381709128672

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