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Properties

Label 2-232-29.6-c1-0-5
Degree $2$
Conductor $232$
Sign $0.996 - 0.0879i$
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  + (2.35 + 0.537i)3-s + (1.76 − 2.21i)5-s + (−0.833 + 3.65i)7-s + (2.54 + 1.22i)9-s + (−1.63 − 3.40i)11-s + (−4.04 + 1.94i)13-s + (5.35 − 4.26i)15-s − 4.98i·17-s + (−2.18 + 0.498i)19-s + (−3.92 + 8.14i)21-s + (5.91 + 7.42i)23-s + (−0.677 − 2.97i)25-s + (−0.329 − 0.262i)27-s + (−5.38 + 0.0249i)29-s + (2.19 + 1.74i)31-s + ⋯
L(s)  = 1  + (1.35 + 0.310i)3-s + (0.790 − 0.991i)5-s + (−0.314 + 1.37i)7-s + (0.848 + 0.408i)9-s + (−0.494 − 1.02i)11-s + (−1.12 + 0.540i)13-s + (1.38 − 1.10i)15-s − 1.21i·17-s + (−0.501 + 0.114i)19-s + (−0.855 + 1.77i)21-s + (1.23 + 1.54i)23-s + (−0.135 − 0.594i)25-s + (−0.0633 − 0.0505i)27-s + (−0.999 + 0.00462i)29-s + (0.393 + 0.313i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.996 - 0.0879i$
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: \((2,\ 232,\ (\ :1/2),\ 0.996 - 0.0879i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86332 + 0.0821329i\)
\(L(\frac12)\) \(\approx\) \(1.86332 + 0.0821329i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.38 - 0.0249i)T \)
good3 \( 1 + (-2.35 - 0.537i)T + (2.70 + 1.30i)T^{2} \)
5 \( 1 + (-1.76 + 2.21i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (0.833 - 3.65i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (1.63 + 3.40i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (4.04 - 1.94i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + 4.98iT - 17T^{2} \)
19 \( 1 + (2.18 - 0.498i)T + (17.1 - 8.24i)T^{2} \)
23 \( 1 + (-5.91 - 7.42i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (-2.19 - 1.74i)T + (6.89 + 30.2i)T^{2} \)
37 \( 1 + (-2.98 + 6.20i)T + (-23.0 - 28.9i)T^{2} \)
41 \( 1 - 2.27iT - 41T^{2} \)
43 \( 1 + (-2.36 + 1.88i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (1.91 + 3.97i)T + (-29.3 + 36.7i)T^{2} \)
53 \( 1 + (2.71 - 3.40i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + 7.79T + 59T^{2} \)
61 \( 1 + (7.09 + 1.61i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (-8.53 - 4.10i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (-1.73 + 0.834i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-0.0751 + 0.0599i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + (2.41 - 5.02i)T + (-49.2 - 61.7i)T^{2} \)
83 \( 1 + (1.49 + 6.53i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-13.1 - 10.5i)T + (19.8 + 86.7i)T^{2} \)
97 \( 1 + (-8.82 + 2.01i)T + (87.3 - 42.0i)T^{2} \)
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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

−12.44326509268197584163083286558, −11.32993734632957916169032882619, −9.600213169244791063386194560319, −9.266477353700800869224305359128, −8.685781793824106161393530551509, −7.50267391672475620025741964253, −5.76929687129053392315749672629, −4.96059694311144418792874135707, −3.14719926684052911575868216231, −2.18724799356826477779131315841, 2.15618333540695372736194795343, 3.10809522160673695978187690055, 4.53116969860570314256310816825, 6.46448401756416148342142177328, 7.28068338751493361207760435988, 8.017013539537454214786144196294, 9.414230407397676734888605584291, 10.24717129186624807531800187737, 10.72239543804904979947849359996, 12.76600605303153221553939805667

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