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Properties

Label 2-315-5.3-c2-0-24
Degree $2$
Conductor $315$
Sign $-0.182 + 0.983i$
Analytic cond. $8.58312$
Root an. cond. $2.92969$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.992 + 0.992i)2-s + 2.02i·4-s + (2.01 − 4.57i)5-s + (1.87 − 1.87i)7-s + (−5.98 − 5.98i)8-s + (2.53 + 6.54i)10-s − 6.89·11-s + (−11.8 − 11.8i)13-s + 3.71i·14-s + 3.77·16-s + (−16.7 + 16.7i)17-s − 8.54i·19-s + (9.27 + 4.09i)20-s + (6.85 − 6.85i)22-s + (−12.4 − 12.4i)23-s + ⋯
L(s)  = 1  + (−0.496 + 0.496i)2-s + 0.507i·4-s + (0.403 − 0.914i)5-s + (0.267 − 0.267i)7-s + (−0.748 − 0.748i)8-s + (0.253 + 0.654i)10-s − 0.627·11-s + (−0.914 − 0.914i)13-s + 0.265i·14-s + 0.235·16-s + (−0.988 + 0.988i)17-s − 0.449i·19-s + (0.463 + 0.204i)20-s + (0.311 − 0.311i)22-s + (−0.542 − 0.542i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.182 + 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.182 + 0.983i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ -0.182 + 0.983i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.355686 - 0.427971i\)
\(L(\frac12)\) \(\approx\) \(0.355686 - 0.427971i\)
\(L(2)\) 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 \( 1 + (-2.01 + 4.57i)T \)
7 \( 1 + (-1.87 + 1.87i)T \)
good2 \( 1 + (0.992 - 0.992i)T - 4iT^{2} \)
11 \( 1 + 6.89T + 121T^{2} \)
13 \( 1 + (11.8 + 11.8i)T + 169iT^{2} \)
17 \( 1 + (16.7 - 16.7i)T - 289iT^{2} \)
19 \( 1 + 8.54iT - 361T^{2} \)
23 \( 1 + (12.4 + 12.4i)T + 529iT^{2} \)
29 \( 1 + 1.33iT - 841T^{2} \)
31 \( 1 + 18.4T + 961T^{2} \)
37 \( 1 + (-31.4 + 31.4i)T - 1.36e3iT^{2} \)
41 \( 1 - 26.7T + 1.68e3T^{2} \)
43 \( 1 + (15.5 + 15.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (-22.1 + 22.1i)T - 2.20e3iT^{2} \)
53 \( 1 + (66.4 + 66.4i)T + 2.80e3iT^{2} \)
59 \( 1 - 81.8iT - 3.48e3T^{2} \)
61 \( 1 + 92.0T + 3.72e3T^{2} \)
67 \( 1 + (-79.2 + 79.2i)T - 4.48e3iT^{2} \)
71 \( 1 - 63.1T + 5.04e3T^{2} \)
73 \( 1 + (-92.9 - 92.9i)T + 5.32e3iT^{2} \)
79 \( 1 + 8.46iT - 6.24e3T^{2} \)
83 \( 1 + (36.2 + 36.2i)T + 6.88e3iT^{2} \)
89 \( 1 - 32.5iT - 7.92e3T^{2} \)
97 \( 1 + (79.2 - 79.2i)T - 9.40e3iT^{2} \)
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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

−11.07360701853061515804435162492, −10.03346978243647567344144244024, −9.115206304135015619991272821216, −8.242011879361798233195307690229, −7.62031949383719432712040992352, −6.38667846052459374418278856394, −5.21204565399548477463700908616, −4.06955598118056157533245119321, −2.37836973642784314783586918296, −0.28894626902085287847331239146, 1.91364636701651218648966937470, 2.78898435131357366693285403498, 4.71585935542852915613067789871, 5.83229192345978449574792882377, 6.84382421189809716323811497556, 7.956096962786862972181996477993, 9.405505537618686543008080867235, 9.674389388281287564337411114815, 10.87817488919411811639470165580, 11.31987091063437894653645486705

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