Properties

Label 2-232-232.11-c1-0-11
Degree $2$
Conductor $232$
Sign $0.479 + 0.877i$
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  + (−0.587 − 1.28i)2-s + (−0.143 − 1.27i)3-s + (−1.31 + 1.51i)4-s + (2.93 + 1.41i)5-s + (−1.55 + 0.931i)6-s + (2.05 + 1.63i)7-s + (2.71 + 0.797i)8-s + (1.32 − 0.302i)9-s + (0.0945 − 4.61i)10-s + (−0.595 − 0.947i)11-s + (2.11 + 1.44i)12-s + (−0.439 + 1.92i)13-s + (0.901 − 3.60i)14-s + (1.37 − 3.93i)15-s + (−0.567 − 3.95i)16-s + (−1.10 + 1.10i)17-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)2-s + (−0.0827 − 0.734i)3-s + (−0.655 + 0.755i)4-s + (1.31 + 0.632i)5-s + (−0.633 + 0.380i)6-s + (0.776 + 0.619i)7-s + (0.959 + 0.282i)8-s + (0.442 − 0.100i)9-s + (0.0298 − 1.45i)10-s + (−0.179 − 0.285i)11-s + (0.609 + 0.418i)12-s + (−0.121 + 0.533i)13-s + (0.240 − 0.964i)14-s + (0.355 − 1.01i)15-s + (−0.141 − 0.989i)16-s + (−0.267 + 0.267i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.479 + 0.877i$
Analytic conductor: \(1.85252\)
Root analytic conductor: \(1.36107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 232,\ (\ :1/2),\ 0.479 + 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04510 - 0.619608i\)
\(L(\frac12)\) \(\approx\) \(1.04510 - 0.619608i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 + 1.28i)T \)
29 \( 1 + (-5.02 + 1.94i)T \)
good3 \( 1 + (0.143 + 1.27i)T + (-2.92 + 0.667i)T^{2} \)
5 \( 1 + (-2.93 - 1.41i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 + (-2.05 - 1.63i)T + (1.55 + 6.82i)T^{2} \)
11 \( 1 + (0.595 + 0.947i)T + (-4.77 + 9.91i)T^{2} \)
13 \( 1 + (0.439 - 1.92i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (1.10 - 1.10i)T - 17iT^{2} \)
19 \( 1 + (5.21 + 0.588i)T + (18.5 + 4.22i)T^{2} \)
23 \( 1 + (1.49 + 3.09i)T + (-14.3 + 17.9i)T^{2} \)
31 \( 1 + (-1.33 - 3.82i)T + (-24.2 + 19.3i)T^{2} \)
37 \( 1 + (9.43 + 5.93i)T + (16.0 + 33.3i)T^{2} \)
41 \( 1 + (-5.66 - 5.66i)T + 41iT^{2} \)
43 \( 1 + (-3.71 + 10.6i)T + (-33.6 - 26.8i)T^{2} \)
47 \( 1 + (-2.88 - 4.58i)T + (-20.3 + 42.3i)T^{2} \)
53 \( 1 + (2.87 - 5.97i)T + (-33.0 - 41.4i)T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 + (4.75 - 0.535i)T + (59.4 - 13.5i)T^{2} \)
67 \( 1 + (-2.51 + 0.573i)T + (60.3 - 29.0i)T^{2} \)
71 \( 1 + (0.174 - 0.766i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (4.03 + 1.41i)T + (57.0 + 45.5i)T^{2} \)
79 \( 1 + (4.55 - 7.24i)T + (-34.2 - 71.1i)T^{2} \)
83 \( 1 + (-1.41 - 1.77i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (12.6 - 4.41i)T + (69.5 - 55.4i)T^{2} \)
97 \( 1 + (-1.45 + 12.9i)T + (-94.5 - 21.5i)T^{2} \)
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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

−12.14423864562486360619683293540, −10.89541359112421018125319563145, −10.31925545522182991329518006803, −9.178290091439080184987303879653, −8.322531323033390285689906146040, −7.02004948366172108105290530674, −5.98124739662119643949612735885, −4.47735155473780999004307341702, −2.49625484403605867149115150781, −1.73449152829675619682577371589, 1.60683004724383457297508052410, 4.41869643041907563660933338908, 5.05796238211486261896643609157, 6.14467358160909326371931581212, 7.42228246411353625023789576816, 8.506382104096567108785904718070, 9.492125175065033624183481459847, 10.19534071258732305235334382798, 10.82794919020411660322184736177, 12.68156158237193814155512201401

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