Properties

Label 2-315-5.2-c2-0-12
Degree $2$
Conductor $315$
Sign $0.913 - 0.407i$
Analytic cond. $8.58312$
Root an. cond. $2.92969$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.01 + 1.01i)2-s − 1.94i·4-s + (−3.97 + 3.03i)5-s + (1.87 + 1.87i)7-s + (6.02 − 6.02i)8-s + (−7.09 − 0.953i)10-s + 10.7·11-s + (15.5 − 15.5i)13-s + 3.78i·14-s + 4.41·16-s + (13.8 + 13.8i)17-s + 33.1i·19-s + (5.90 + 7.74i)20-s + (10.8 + 10.8i)22-s + (−8.39 + 8.39i)23-s + ⋯
L(s)  = 1  + (0.506 + 0.506i)2-s − 0.487i·4-s + (−0.794 + 0.606i)5-s + (0.267 + 0.267i)7-s + (0.753 − 0.753i)8-s + (−0.709 − 0.0953i)10-s + 0.973·11-s + (1.19 − 1.19i)13-s + 0.270i·14-s + 0.275·16-s + (0.816 + 0.816i)17-s + 1.74i·19-s + (0.295 + 0.387i)20-s + (0.493 + 0.493i)22-s + (−0.364 + 0.364i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.913 - 0.407i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ 0.913 - 0.407i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.13950 + 0.456148i\)
\(L(\frac12)\) \(\approx\) \(2.13950 + 0.456148i\)
\(L(2)\) 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 \( 1 + (3.97 - 3.03i)T \)
7 \( 1 + (-1.87 - 1.87i)T \)
good2 \( 1 + (-1.01 - 1.01i)T + 4iT^{2} \)
11 \( 1 - 10.7T + 121T^{2} \)
13 \( 1 + (-15.5 + 15.5i)T - 169iT^{2} \)
17 \( 1 + (-13.8 - 13.8i)T + 289iT^{2} \)
19 \( 1 - 33.1iT - 361T^{2} \)
23 \( 1 + (8.39 - 8.39i)T - 529iT^{2} \)
29 \( 1 + 42.0iT - 841T^{2} \)
31 \( 1 - 44.7T + 961T^{2} \)
37 \( 1 + (-21.6 - 21.6i)T + 1.36e3iT^{2} \)
41 \( 1 - 22.6T + 1.68e3T^{2} \)
43 \( 1 + (-9.41 + 9.41i)T - 1.84e3iT^{2} \)
47 \( 1 + (46.0 + 46.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (51.4 - 51.4i)T - 2.80e3iT^{2} \)
59 \( 1 - 5.55iT - 3.48e3T^{2} \)
61 \( 1 + 49.2T + 3.72e3T^{2} \)
67 \( 1 + (38.8 + 38.8i)T + 4.48e3iT^{2} \)
71 \( 1 + 23.5T + 5.04e3T^{2} \)
73 \( 1 + (60.1 - 60.1i)T - 5.32e3iT^{2} \)
79 \( 1 + 8.04iT - 6.24e3T^{2} \)
83 \( 1 + (-5.76 + 5.76i)T - 6.88e3iT^{2} \)
89 \( 1 + 62.2iT - 7.92e3T^{2} \)
97 \( 1 + (-21.9 - 21.9i)T + 9.40e3iT^{2} \)
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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

−11.56629824422905983816306939064, −10.52367888061992580438229597508, −9.901491112421208329820305025387, −8.273603992343980266874299738256, −7.72737189832690036467705206402, −6.19747650556867953716060023853, −5.93972490891020600467941052558, −4.29305155097585527270613536635, −3.45310699194323208549370524398, −1.26974205364870616045069282506, 1.26841171804661192551213537055, 3.11983340985087312673194970095, 4.21041825716059410481822482176, 4.81249714304843639930092763940, 6.59717853304144298895192740157, 7.58532094521362374590610025550, 8.624117049612320896661722741280, 9.283177633061580674489267391325, 11.02192118132760773274331472600, 11.48092602901458489471691467441

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