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Properties

Label 2-315-105.104-c1-0-8
Degree $2$
Conductor $315$
Sign $0.565 + 0.824i$
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  − 2.33·2-s + 3.44·4-s + (1.33 − 1.79i)5-s + (2.53 − 0.741i)7-s − 3.38·8-s + (−3.11 + 4.19i)10-s − 1.41i·11-s − 1.14·13-s + (−5.92 + 1.73i)14-s + 1.00·16-s + 5.20i·17-s − 4.61i·19-s + (4.59 − 6.19i)20-s + 3.30i·22-s + 3.61·23-s + ⋯
L(s)  = 1  − 1.65·2-s + 1.72·4-s + (0.595 − 0.803i)5-s + (0.959 − 0.280i)7-s − 1.19·8-s + (−0.983 + 1.32i)10-s − 0.426i·11-s − 0.316·13-s + (−1.58 + 0.462i)14-s + 0.250·16-s + 1.26i·17-s − 1.05i·19-s + (1.02 − 1.38i)20-s + 0.703i·22-s + 0.754·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.565 + 0.824i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (314, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.565 + 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.635601 - 0.334776i\)
\(L(\frac12)\) \(\approx\) \(0.635601 - 0.334776i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.33 + 1.79i)T \)
7 \( 1 + (-2.53 + 0.741i)T \)
good2 \( 1 + 2.33T + 2T^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + 1.14T + 13T^{2} \)
17 \( 1 - 5.20iT - 17T^{2} \)
19 \( 1 + 4.61iT - 19T^{2} \)
23 \( 1 - 3.61T + 23T^{2} \)
29 \( 1 + 8.34iT - 29T^{2} \)
31 \( 1 - 8.38iT - 31T^{2} \)
37 \( 1 + 8.08iT - 37T^{2} \)
41 \( 1 - 9.19T + 41T^{2} \)
43 \( 1 + 5.11iT - 43T^{2} \)
47 \( 1 + 1.61iT - 47T^{2} \)
53 \( 1 + 3.14T + 53T^{2} \)
59 \( 1 + 11.8T + 59T^{2} \)
61 \( 1 - 7.53iT - 61T^{2} \)
67 \( 1 - 9.57iT - 67T^{2} \)
71 \( 1 - 5.51iT - 71T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 1.61iT - 83T^{2} \)
89 \( 1 - 7.99T + 89T^{2} \)
97 \( 1 + 1.14T + 97T^{2} \)
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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

−11.05236444542862795083373696980, −10.52768306617244969838279622123, −9.434457561813860887711457211274, −8.734418456982292466817100571064, −8.063265175788738414402971272405, −7.05330488408227811957843979761, −5.76238228065980783396427988691, −4.47354906631894105820707170276, −2.22236277286238987950083540960, −0.985339130326871560827260802555, 1.58502097505968508229793693947, 2.75905827673759531140957856664, 4.93891356838116099712119646456, 6.32529429214864168851864823806, 7.37747119444956771199326920994, 7.965448087487572606369120706620, 9.245622096244019810732435888892, 9.707012063148060181207851765774, 10.79252167238518548785165939931, 11.26583346441203810286585502441

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