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Properties

Label 2-945-315.52-c1-0-39
Degree $2$
Conductor $945$
Sign $-0.658 + 0.752i$
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  + (0.0825 + 0.0825i)2-s − 1.98i·4-s + (−1.53 − 1.62i)5-s + (2.61 − 0.400i)7-s + (0.329 − 0.329i)8-s + (0.00682 − 0.261i)10-s + (−0.455 − 0.789i)11-s + (1.53 − 5.74i)13-s + (0.249 + 0.182i)14-s − 3.91·16-s + (0.884 + 3.29i)17-s + (−0.815 − 1.41i)19-s + (−3.22 + 3.05i)20-s + (0.0275 − 0.102i)22-s + (0.844 + 3.15i)23-s + ⋯
L(s)  = 1  + (0.0583 + 0.0583i)2-s − 0.993i·4-s + (−0.688 − 0.725i)5-s + (0.988 − 0.151i)7-s + (0.116 − 0.116i)8-s + (0.00215 − 0.0825i)10-s + (−0.137 − 0.238i)11-s + (0.426 − 1.59i)13-s + (0.0665 + 0.0488i)14-s − 0.979·16-s + (0.214 + 0.800i)17-s + (−0.187 − 0.323i)19-s + (−0.720 + 0.683i)20-s + (0.00587 − 0.0219i)22-s + (0.176 + 0.657i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.658 + 0.752i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ -0.658 + 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.554238 - 1.22133i\)
\(L(\frac12)\) \(\approx\) \(0.554238 - 1.22133i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.53 + 1.62i)T \)
7 \( 1 + (-2.61 + 0.400i)T \)
good2 \( 1 + (-0.0825 - 0.0825i)T + 2iT^{2} \)
11 \( 1 + (0.455 + 0.789i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.53 + 5.74i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-0.884 - 3.29i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.815 + 1.41i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.844 - 3.15i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (5.74 + 3.31i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.16iT - 31T^{2} \)
37 \( 1 + (0.935 - 3.49i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-5.95 + 3.43i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.11 + 4.14i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (4.13 - 4.13i)T - 47iT^{2} \)
53 \( 1 + (3.64 + 13.5i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + 4.13T + 59T^{2} \)
61 \( 1 - 7.18iT - 61T^{2} \)
67 \( 1 + (10.7 + 10.7i)T + 67iT^{2} \)
71 \( 1 + 7.48T + 71T^{2} \)
73 \( 1 + (-3.70 + 0.993i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 - 9.14iT - 79T^{2} \)
83 \( 1 + (-10.7 + 2.88i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-1.92 - 3.33i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.83 - 6.83i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.792837295986688615746576575418, −8.799886661531483751903619229378, −8.069628364134015224027702585133, −7.43216398843352723130596870167, −5.99663649744953299739017070279, −5.37170209245611295822453327677, −4.58323838624074548734979480430, −3.49946001227090685355370278609, −1.75363013456440769602632688934, −0.63150433888425499284887721934, 1.94518410433909645493652454679, 3.06879605631511713521093312022, 4.14046782175424680038262117331, 4.73400969671503898640305624721, 6.26526924243217225004947766705, 7.25615437814512410976899001661, 7.66232987052641540712212593597, 8.638003721817354166356880117856, 9.273430530246825750788079491458, 10.66618242719411485199944735375

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