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Properties

Label 2-315-21.20-c9-0-31
Degree $2$
Conductor $315$
Sign $-0.997 - 0.0722i$
Analytic cond. $162.236$
Root an. cond. $12.7372$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16.2i·2-s + 246.·4-s + 625·5-s + (−4.03e3 + 4.90e3i)7-s + 1.23e4i·8-s + 1.01e4i·10-s + 7.41e4i·11-s − 1.88e5i·13-s + (−7.99e4 − 6.56e4i)14-s − 7.47e4·16-s + 4.57e5·17-s + 5.34e5i·19-s + 1.54e5·20-s − 1.20e6·22-s + 8.58e5i·23-s + ⋯
L(s)  = 1  + 0.719i·2-s + 0.482·4-s + 0.447·5-s + (−0.634 + 0.772i)7-s + 1.06i·8-s + 0.321i·10-s + 1.52i·11-s − 1.83i·13-s + (−0.556 − 0.456i)14-s − 0.285·16-s + 1.32·17-s + 0.940i·19-s + 0.215·20-s − 1.09·22-s + 0.639i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0722i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.997 - 0.0722i$
Analytic conductor: \(162.236\)
Root analytic conductor: \(12.7372\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :9/2),\ -0.997 - 0.0722i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.507157274\)
\(L(\frac12)\) \(\approx\) \(2.507157274\)
\(L(\frac{11}{2})\) 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 \( 1 - 625T \)
7 \( 1 + (4.03e3 - 4.90e3i)T \)
good2 \( 1 - 16.2iT - 512T^{2} \)
11 \( 1 - 7.41e4iT - 2.35e9T^{2} \)
13 \( 1 + 1.88e5iT - 1.06e10T^{2} \)
17 \( 1 - 4.57e5T + 1.18e11T^{2} \)
19 \( 1 - 5.34e5iT - 3.22e11T^{2} \)
23 \( 1 - 8.58e5iT - 1.80e12T^{2} \)
29 \( 1 - 4.72e6iT - 1.45e13T^{2} \)
31 \( 1 - 2.47e5iT - 2.64e13T^{2} \)
37 \( 1 - 1.57e7T + 1.29e14T^{2} \)
41 \( 1 + 1.50e7T + 3.27e14T^{2} \)
43 \( 1 - 4.15e6T + 5.02e14T^{2} \)
47 \( 1 + 1.30e7T + 1.11e15T^{2} \)
53 \( 1 + 9.25e7iT - 3.29e15T^{2} \)
59 \( 1 - 1.12e8T + 8.66e15T^{2} \)
61 \( 1 + 3.91e7iT - 1.16e16T^{2} \)
67 \( 1 - 3.12e6T + 2.72e16T^{2} \)
71 \( 1 - 3.51e8iT - 4.58e16T^{2} \)
73 \( 1 - 4.73e8iT - 5.88e16T^{2} \)
79 \( 1 + 3.62e8T + 1.19e17T^{2} \)
83 \( 1 + 2.56e8T + 1.86e17T^{2} \)
89 \( 1 - 7.32e8T + 3.50e17T^{2} \)
97 \( 1 + 2.33e8iT - 7.60e17T^{2} \)
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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

−10.14471516472393140718120188782, −9.873990463128710952078723463920, −8.412477134349977546943855874851, −7.63119292742937022006848996882, −6.74524648077229300560143729261, −5.61192559520140008683345926448, −5.30650312979133634960231573400, −3.37357298102064781164578246180, −2.45036850243422958145537490433, −1.33686365042867680018281314550, 0.45923721410385796047079988224, 1.25800571251061845940148500776, 2.49875105321338939995313811091, 3.38578850722041178655770119295, 4.35467374873729417454261812019, 6.03647217165897199244409674808, 6.59003200469797116757978803884, 7.64906143520517208804001337968, 9.044568049559795365344686475803, 9.789291157330055211239984523169

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