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Properties

Label 2-232-232.109-c1-0-16
Degree $2$
Conductor $232$
Sign $0.999 - 0.0314i$
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  + (−1.28 − 0.583i)2-s + (2.14 + 1.03i)3-s + (1.31 + 1.50i)4-s + (3.03 − 0.692i)5-s + (−2.16 − 2.58i)6-s + (0.547 + 0.263i)7-s + (−0.822 − 2.70i)8-s + (1.67 + 2.10i)9-s + (−4.31 − 0.878i)10-s + (−1.20 + 1.51i)11-s + (1.27 + 4.59i)12-s + (−4.76 − 3.79i)13-s + (−0.551 − 0.658i)14-s + (7.24 + 1.65i)15-s + (−0.518 + 3.96i)16-s + 1.48i·17-s + ⋯
L(s)  = 1  + (−0.910 − 0.412i)2-s + (1.24 + 0.597i)3-s + (0.659 + 0.751i)4-s + (1.35 − 0.309i)5-s + (−0.883 − 1.05i)6-s + (0.206 + 0.0995i)7-s + (−0.290 − 0.956i)8-s + (0.558 + 0.700i)9-s + (−1.36 − 0.277i)10-s + (−0.364 + 0.457i)11-s + (0.369 + 1.32i)12-s + (−1.32 − 1.05i)13-s + (−0.147 − 0.176i)14-s + (1.86 + 0.426i)15-s + (−0.129 + 0.991i)16-s + 0.360i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.999 - 0.0314i$
Analytic conductor: \(1.85252\)
Root analytic conductor: \(1.36107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 232,\ (\ :1/2),\ 0.999 - 0.0314i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38818 + 0.0218525i\)
\(L(\frac12)\) \(\approx\) \(1.38818 + 0.0218525i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 + 0.583i)T \)
29 \( 1 + (-4.72 + 2.58i)T \)
good3 \( 1 + (-2.14 - 1.03i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (-3.03 + 0.692i)T + (4.50 - 2.16i)T^{2} \)
7 \( 1 + (-0.547 - 0.263i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (1.20 - 1.51i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (4.76 + 3.79i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 - 1.48iT - 17T^{2} \)
19 \( 1 + (5.08 - 2.44i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (-0.367 + 1.61i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (-2.09 + 0.478i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (3.22 + 4.03i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 - 4.90iT - 41T^{2} \)
43 \( 1 + (1.95 - 8.55i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-9.25 - 7.38i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (7.00 - 1.59i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 - 7.85iT - 59T^{2} \)
61 \( 1 + (10.2 + 4.95i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (-10.2 + 8.16i)T + (14.9 - 65.3i)T^{2} \)
71 \( 1 + (-1.20 + 1.50i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-10.2 - 2.34i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + (6.98 - 5.56i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (5.24 + 10.8i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (-6.57 + 1.50i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (2.45 + 5.10i)T + (-60.4 + 75.8i)T^{2} \)
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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

−12.39120636804891455627722165923, −10.65000372039467214164466418618, −9.965055209061849186560627883717, −9.488594326415365244393313981449, −8.457832024956177138373720680169, −7.76290827408488083184175954088, −6.18483741793305218426688514715, −4.60254738719069560209411085152, −2.88279146972987551263219635404, −2.05706400561086915806271467779, 1.92053993606849047991831756639, 2.63514384963778515078333522240, 5.14623992518003702685658561486, 6.56906417545901511017990849183, 7.21511481118320975155342435483, 8.407156794251208552198111360082, 9.122960853675772774898812350825, 9.917532387374838861785220480916, 10.84684979672481946955667981605, 12.24490328230496070357722094193

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