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Properties

Label 2-315-63.59-c1-0-4
Degree $2$
Conductor $315$
Sign $-0.307 - 0.951i$
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  + (−0.0228 − 0.0132i)2-s + (−0.396 + 1.68i)3-s + (−0.999 − 1.73i)4-s + 5-s + (0.0313 − 0.0333i)6-s + (−1.43 + 2.22i)7-s + 0.105i·8-s + (−2.68 − 1.33i)9-s + (−0.0228 − 0.0132i)10-s + 6.31i·11-s + (3.31 − 0.998i)12-s + (2.98 + 1.72i)13-s + (0.0621 − 0.0318i)14-s + (−0.396 + 1.68i)15-s + (−1.99 + 3.46i)16-s + (1.13 − 1.96i)17-s + ⋯
L(s)  = 1  + (−0.0161 − 0.00933i)2-s + (−0.229 + 0.973i)3-s + (−0.499 − 0.865i)4-s + 0.447·5-s + (0.0127 − 0.0135i)6-s + (−0.542 + 0.840i)7-s + 0.0373i·8-s + (−0.894 − 0.446i)9-s + (−0.00723 − 0.00417i)10-s + 1.90i·11-s + (0.957 − 0.288i)12-s + (0.828 + 0.478i)13-s + (0.0166 − 0.00852i)14-s + (−0.102 + 0.435i)15-s + (−0.499 + 0.865i)16-s + (0.274 − 0.475i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.307 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.307 - 0.951i$
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: \((2,\ 315,\ (\ :1/2),\ -0.307 - 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.529235 + 0.727460i\)
\(L(\frac12)\) \(\approx\) \(0.529235 + 0.727460i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.396 - 1.68i)T \)
5 \( 1 - T \)
7 \( 1 + (1.43 - 2.22i)T \)
good2 \( 1 + (0.0228 + 0.0132i)T + (1 + 1.73i)T^{2} \)
11 \( 1 - 6.31iT - 11T^{2} \)
13 \( 1 + (-2.98 - 1.72i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.13 + 1.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.32 - 2.49i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.630iT - 23T^{2} \)
29 \( 1 + (-1.23 + 0.713i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.76 - 4.48i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.69 - 8.12i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.66 + 8.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.21 + 2.10i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.24 + 9.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.53 - 1.46i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.136 + 0.236i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.11 - 2.95i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.942 - 1.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.59iT - 71T^{2} \)
73 \( 1 + (0.0467 + 0.0270i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.98 + 6.89i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.66 - 8.08i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.0707 - 0.122i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.37 - 0.791i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.90810597283270734050198077300, −10.66483552797413102050850733329, −10.00184012195723889080440400547, −9.332314830090367924276131866582, −8.684103287673211537995936870684, −6.77618059220058525329077431605, −5.81821618045401017668679319946, −4.99061017700602480893855491757, −3.94687034377022944704843443498, −2.08570504154422048129938675891, 0.68432620401927672367768476542, 2.87512693289227456455638859682, 3.93794665166817790633432837654, 5.72418406919646079622093099792, 6.44239390299729584873147563978, 7.65808363237386748971396335070, 8.392141876640683407134937077742, 9.236928880449170351504588945995, 10.83113899464519314196082641289, 11.23077293183686657310077319185

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