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Properties

Label 2-990-11.5-c1-0-11
Degree $2$
Conductor $990$
Sign $0.957 + 0.288i$
Analytic cond. $7.90518$
Root an. cond. $2.81161$
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  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)5-s + (−0.809 − 0.587i)8-s + 0.999·10-s + (2.54 − 2.12i)11-s + (−1.11 − 3.44i)13-s + (0.309 − 0.951i)16-s + (0.190 − 0.587i)17-s + (−1.61 − 1.17i)19-s + (0.309 + 0.951i)20-s + (2.80 + 1.76i)22-s + 1.85·23-s + (−0.809 − 0.587i)25-s + (2.92 − 2.12i)26-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.404 + 0.293i)4-s + (0.138 − 0.425i)5-s + (−0.286 − 0.207i)8-s + 0.316·10-s + (0.767 − 0.641i)11-s + (−0.310 − 0.954i)13-s + (0.0772 − 0.237i)16-s + (0.0463 − 0.142i)17-s + (−0.371 − 0.269i)19-s + (0.0690 + 0.212i)20-s + (0.598 + 0.375i)22-s + 0.386·23-s + (−0.161 − 0.117i)25-s + (0.574 − 0.417i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.957 + 0.288i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ 0.957 + 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62885 - 0.239660i\)
\(L(\frac12)\) \(\approx\) \(1.62885 - 0.239660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 \)
5 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-2.54 + 2.12i)T \)
good7 \( 1 + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.11 + 3.44i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.190 + 0.587i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.61 + 1.17i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 1.85T + 23T^{2} \)
29 \( 1 + (-3.54 + 2.57i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.35 + 10.3i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.118 + 0.0857i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-7.85 - 5.70i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 3.85T + 43T^{2} \)
47 \( 1 + (-8.16 - 5.93i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.14 + 6.60i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-6.35 + 4.61i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.76 + 5.42i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 + (-2.14 + 6.60i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.281 - 0.865i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.61 - 4.97i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 4.47T + 89T^{2} \)
97 \( 1 + (-1.47 - 4.53i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.642752767376433007472038021066, −9.112153999944820851749673351761, −8.135402600357292182732506216062, −7.53959964377747684025275039157, −6.34859534061095635240282168534, −5.78911371402618603315384344224, −4.76443999855550186828502120109, −3.86654615883181816840702878874, −2.62968408974280825859323671820, −0.76828585891427704041847145933, 1.47365803245381312926306666952, 2.53526591237601285627317162460, 3.75111592659723812714565320994, 4.53619947349174470798499809402, 5.61715179263608508980609904255, 6.71801833622637522130213915188, 7.29072089498054440954754353497, 8.747701208241430809884253211394, 9.199600246061414224731726967036, 10.23560802769395126164791861800

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