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Properties

Label 2-315-63.59-c1-0-1
Degree $2$
Conductor $315$
Sign $0.264 - 0.964i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
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  + (−2.43 − 1.40i)2-s + (−1.45 + 0.939i)3-s + (2.94 + 5.09i)4-s + 5-s + (4.85 − 0.240i)6-s + (0.979 + 2.45i)7-s − 10.9i·8-s + (1.23 − 2.73i)9-s + (−2.43 − 1.40i)10-s + 2.94i·11-s + (−9.06 − 4.65i)12-s + (−1.90 − 1.10i)13-s + (1.06 − 7.35i)14-s + (−1.45 + 0.939i)15-s + (−9.41 + 16.3i)16-s + (−1.11 + 1.93i)17-s + ⋯
L(s)  = 1  + (−1.71 − 0.992i)2-s + (−0.840 + 0.542i)3-s + (1.47 + 2.54i)4-s + 0.447·5-s + (1.98 − 0.0983i)6-s + (0.370 + 0.928i)7-s − 3.85i·8-s + (0.411 − 0.911i)9-s + (−0.768 − 0.443i)10-s + 0.889i·11-s + (−2.61 − 1.34i)12-s + (−0.529 − 0.305i)13-s + (0.285 − 1.96i)14-s + (−0.375 + 0.242i)15-s + (−2.35 + 4.07i)16-s + (−0.270 + 0.468i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.264 - 0.964i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.264 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.303121 + 0.231186i\)
\(L(\frac12)\) \(\approx\) \(0.303121 + 0.231186i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.45 - 0.939i)T \)
5 \( 1 - T \)
7 \( 1 + (-0.979 - 2.45i)T \)
good2 \( 1 + (2.43 + 1.40i)T + (1 + 1.73i)T^{2} \)
11 \( 1 - 2.94iT - 11T^{2} \)
13 \( 1 + (1.90 + 1.10i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.11 - 1.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.10 + 1.21i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.164iT - 23T^{2} \)
29 \( 1 + (1.75 - 1.01i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.43 - 0.825i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.761 - 1.31i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.32 - 5.75i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.57 - 7.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.97 - 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.54 + 1.47i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.12 + 1.94i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.17 + 3.56i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.68 - 2.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.49iT - 71T^{2} \)
73 \( 1 + (-3.71 - 2.14i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.143 - 0.248i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.67 + 4.62i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.937 + 1.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.0 - 5.80i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.44099776165545482846807595351, −10.90818616543361493086948707684, −9.717087953372644736314919309086, −9.585630259724498336884213350990, −8.430020831247043105934913455197, −7.33158753635788606281758966362, −6.20842214721569845436784638262, −4.64089697379742158258061624380, −2.96183008482469123633262869633, −1.60852248485896165920389856339, 0.54047600300512597397684399281, 1.89874348590798932228595806880, 5.06733757327857143657914885892, 5.92842629878913842104981525294, 6.93131593767093155829664185682, 7.48598454463203606880967809292, 8.473356608712326125169886425629, 9.566164056025293590481225676548, 10.44449491862298320343081364835, 11.04680082054106091596100855620

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