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Properties

Label 2-322-161.103-c1-0-1
Degree $2$
Conductor $322$
Sign $-0.192 - 0.981i$
Analytic cond. $2.57118$
Root an. cond. $1.60349$
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  + (−0.0475 − 0.998i)2-s + (1.18 + 2.96i)3-s + (−0.995 + 0.0950i)4-s + (−0.274 + 1.12i)5-s + (2.90 − 1.32i)6-s + (−2.55 + 0.669i)7-s + (0.142 + 0.989i)8-s + (−5.21 + 4.97i)9-s + (1.14 + 0.219i)10-s + (0.728 + 0.0347i)11-s + (−1.46 − 2.84i)12-s + (−0.469 − 0.406i)13-s + (0.790 + 2.52i)14-s + (−3.67 + 0.528i)15-s + (0.981 − 0.189i)16-s + (2.06 − 2.90i)17-s + ⋯
L(s)  = 1  + (−0.0336 − 0.706i)2-s + (0.685 + 1.71i)3-s + (−0.497 + 0.0475i)4-s + (−0.122 + 0.505i)5-s + (1.18 − 0.541i)6-s + (−0.967 + 0.253i)7-s + (0.0503 + 0.349i)8-s + (−1.73 + 1.65i)9-s + (0.360 + 0.0695i)10-s + (0.219 + 0.0104i)11-s + (−0.422 − 0.819i)12-s + (−0.130 − 0.112i)13-s + (0.211 + 0.674i)14-s + (−0.949 + 0.136i)15-s + (0.245 − 0.0473i)16-s + (0.501 − 0.703i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.192 - 0.981i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ -0.192 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.770068 + 0.935685i\)
\(L(\frac12)\) \(\approx\) \(0.770068 + 0.935685i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0475 + 0.998i)T \)
7 \( 1 + (2.55 - 0.669i)T \)
23 \( 1 + (0.461 - 4.77i)T \)
good3 \( 1 + (-1.18 - 2.96i)T + (-2.17 + 2.07i)T^{2} \)
5 \( 1 + (0.274 - 1.12i)T + (-4.44 - 2.29i)T^{2} \)
11 \( 1 + (-0.728 - 0.0347i)T + (10.9 + 1.04i)T^{2} \)
13 \( 1 + (0.469 + 0.406i)T + (1.85 + 12.8i)T^{2} \)
17 \( 1 + (-2.06 + 2.90i)T + (-5.56 - 16.0i)T^{2} \)
19 \( 1 + (1.39 + 1.95i)T + (-6.21 + 17.9i)T^{2} \)
29 \( 1 + (-2.46 - 5.39i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (-5.86 - 7.45i)T + (-7.30 + 30.1i)T^{2} \)
37 \( 1 + (0.347 + 0.364i)T + (-1.76 + 36.9i)T^{2} \)
41 \( 1 + (2.57 + 8.78i)T + (-34.4 + 22.1i)T^{2} \)
43 \( 1 + (-10.6 - 1.53i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + (-6.07 - 3.50i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.82 + 0.633i)T + (41.6 + 32.7i)T^{2} \)
59 \( 1 + (-0.237 + 1.23i)T + (-54.7 - 21.9i)T^{2} \)
61 \( 1 + (0.813 + 0.325i)T + (44.1 + 42.0i)T^{2} \)
67 \( 1 + (3.39 - 6.57i)T + (-38.8 - 54.5i)T^{2} \)
71 \( 1 + (5.42 - 3.48i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (0.992 + 10.3i)T + (-71.6 + 13.8i)T^{2} \)
79 \( 1 + (1.34 - 0.467i)T + (62.0 - 48.8i)T^{2} \)
83 \( 1 + (8.40 + 2.46i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (11.5 + 9.05i)T + (20.9 + 86.4i)T^{2} \)
97 \( 1 + (-17.0 + 5.01i)T + (81.6 - 52.4i)T^{2} \)
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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

−11.61526480880597567496968386123, −10.61323221792309748975516453432, −10.14440137948759322449187402237, −9.209916466522794966672774534811, −8.753377745912742833094742931965, −7.22732727692263176788197982586, −5.58613515761250162499415610020, −4.50526107665866979031582575791, −3.32931285092843339569586913007, −2.84066871613244680014304548785, 0.838023138548006226522433443664, 2.62737791039574555729171732269, 4.12278470215835603622868542658, 6.06117978121669016736126639643, 6.49249667125111499803169603525, 7.58888992735260586923688732430, 8.262715717049062691533423350476, 9.063375812854775793538891557344, 10.16386465214934018207185305319, 11.88996816663214404939947955405

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