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Properties

Label 2-315-9.4-c1-0-21
Degree $2$
Conductor $315$
Sign $0.0417 + 0.999i$
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  + (0.368 + 0.638i)2-s + (−0.764 − 1.55i)3-s + (0.728 − 1.26i)4-s + (0.5 − 0.866i)5-s + (0.710 − 1.06i)6-s + (−0.5 − 0.866i)7-s + 2.54·8-s + (−1.83 + 2.37i)9-s + 0.737·10-s + (−3.17 − 5.49i)11-s + (−2.51 − 0.167i)12-s + (−2.94 + 5.10i)13-s + (0.368 − 0.638i)14-s + (−1.72 − 0.114i)15-s + (−0.517 − 0.895i)16-s + 5.01·17-s + ⋯
L(s)  = 1  + (0.260 + 0.451i)2-s + (−0.441 − 0.897i)3-s + (0.364 − 0.630i)4-s + (0.223 − 0.387i)5-s + (0.290 − 0.433i)6-s + (−0.188 − 0.327i)7-s + 0.900·8-s + (−0.610 + 0.792i)9-s + 0.233·10-s + (−0.957 − 1.65i)11-s + (−0.726 − 0.0483i)12-s + (−0.817 + 1.41i)13-s + (0.0985 − 0.170i)14-s + (−0.446 − 0.0296i)15-s + (−0.129 − 0.223i)16-s + 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0417 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0417 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.0417 + 0.999i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.0417 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.963155 - 0.923784i\)
\(L(\frac12)\) \(\approx\) \(0.963155 - 0.923784i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.764 + 1.55i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.368 - 0.638i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (3.17 + 5.49i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.94 - 5.10i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.01T + 17T^{2} \)
19 \( 1 + 0.158T + 19T^{2} \)
23 \( 1 + (-3.10 + 5.38i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.235 + 0.407i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.660 - 1.14i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.21T + 37T^{2} \)
41 \( 1 + (1.06 - 1.84i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.32 - 2.29i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.47 - 7.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.52T + 53T^{2} \)
59 \( 1 + (-3.09 + 5.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.284 - 0.492i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.27 - 9.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.75T + 71T^{2} \)
73 \( 1 - 1.95T + 73T^{2} \)
79 \( 1 + (-3.45 - 5.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.17 - 3.75i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.25T + 89T^{2} \)
97 \( 1 + (2.35 + 4.07i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.36744562293690933632205562146, −10.74949450930686479879302377455, −9.632911703667033367504849360044, −8.287346435824175605039215283350, −7.39265478277647961353680279406, −6.40946949403562223598422718810, −5.68893606681514920959815629273, −4.73545825699800088425528836830, −2.59597993187063412911527148542, −0.967018807033512020579233366939, 2.50439048061159417388580620427, 3.43078664060477836966084675909, 4.82914638160769928665313319346, 5.63176525099746485931229789055, 7.21983630035239540586638978973, 7.88844811043013778256980744716, 9.553472679811281514695636230625, 10.17000258078803241720967880983, 10.84595663618110506110582885869, 12.00965043804172786707126953175

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