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Properties

Label 2-945-105.53-c1-0-13
Degree $2$
Conductor $945$
Sign $-0.786 - 0.618i$
Analytic cond. $7.54586$
Root an. cond. $2.74697$
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  + (−0.136 + 0.509i)2-s + (1.49 + 0.860i)4-s + (−1.13 + 1.92i)5-s + (0.574 + 2.58i)7-s + (−1.38 + 1.38i)8-s + (−0.829 − 0.840i)10-s + (3.52 + 2.03i)11-s + (−1.23 − 1.23i)13-s + (−1.39 − 0.0601i)14-s + (1.20 + 2.08i)16-s + (−5.70 + 1.52i)17-s + (6.15 − 3.55i)19-s + (−3.34 + 1.90i)20-s + (−1.51 + 1.51i)22-s + (1.28 + 0.344i)23-s + ⋯
L(s)  = 1  + (−0.0966 + 0.360i)2-s + (0.745 + 0.430i)4-s + (−0.505 + 0.862i)5-s + (0.216 + 0.976i)7-s + (−0.491 + 0.491i)8-s + (−0.262 − 0.265i)10-s + (1.06 + 0.612i)11-s + (−0.343 − 0.343i)13-s + (−0.372 − 0.0160i)14-s + (0.300 + 0.520i)16-s + (−1.38 + 0.370i)17-s + (1.41 − 0.814i)19-s + (−0.748 + 0.425i)20-s + (−0.323 + 0.323i)22-s + (0.268 + 0.0718i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 - 0.618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.786 - 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.786 - 0.618i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ -0.786 - 0.618i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.509944 + 1.47343i\)
\(L(\frac12)\) \(\approx\) \(0.509944 + 1.47343i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (1.13 - 1.92i)T \)
7 \( 1 + (-0.574 - 2.58i)T \)
good2 \( 1 + (0.136 - 0.509i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-3.52 - 2.03i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.23 + 1.23i)T + 13iT^{2} \)
17 \( 1 + (5.70 - 1.52i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-6.15 + 3.55i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.28 - 0.344i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 3.96T + 29T^{2} \)
31 \( 1 + (1.17 - 2.03i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.00 + 1.07i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 6.21iT - 41T^{2} \)
43 \( 1 + (-6.67 - 6.67i)T + 43iT^{2} \)
47 \( 1 + (-3.51 + 13.1i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.23 - 8.35i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.03 - 6.99i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.0936 - 0.162i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.130 + 0.485i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 12.8iT - 71T^{2} \)
73 \( 1 + (1.33 - 0.357i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-13.1 + 7.61i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.35 + 8.35i)T - 83iT^{2} \)
89 \( 1 + (1.12 + 1.95i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.99 - 7.99i)T - 97iT^{2} \)
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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

−10.57771241508711394744607894861, −9.271980388162009002829748682140, −8.741890138544554120755703151943, −7.57222643755064590479964311576, −7.08695104917363517744011402211, −6.33870213071076393192155889800, −5.32986625847173348151972379865, −4.00008044083784043948686942442, −2.92247432158753830839969508096, −2.05647879199352749102970976909, 0.75144344429831670215689946610, 1.75189892166532535533367043136, 3.37777864236262476854185670593, 4.22406093189802130895008340160, 5.28058283118519782939853701181, 6.39304430922541390622831273092, 7.19850876850212016983039488860, 7.955245230444538502776661654592, 9.199866120260492217444866521523, 9.551515356926031684590300195107

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