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Properties

Label 2-315-35.3-c1-0-13
Degree $2$
Conductor $315$
Sign $0.543 + 0.839i$
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  + (0.969 + 0.259i)2-s + (−0.859 − 0.496i)4-s + (−0.803 − 2.08i)5-s + (2.42 + 1.06i)7-s + (−2.12 − 2.12i)8-s + (−0.237 − 2.23i)10-s + (1.78 − 3.08i)11-s + (2.78 − 2.78i)13-s + (2.07 + 1.65i)14-s + (−0.514 − 0.891i)16-s + (0.506 − 0.135i)17-s + (−2.06 − 3.57i)19-s + (−0.344 + 2.19i)20-s + (2.53 − 2.53i)22-s + (−0.668 + 2.49i)23-s + ⋯
L(s)  = 1  + (0.685 + 0.183i)2-s + (−0.429 − 0.248i)4-s + (−0.359 − 0.933i)5-s + (0.915 + 0.401i)7-s + (−0.750 − 0.750i)8-s + (−0.0750 − 0.705i)10-s + (0.537 − 0.931i)11-s + (0.772 − 0.772i)13-s + (0.554 + 0.443i)14-s + (−0.128 − 0.222i)16-s + (0.122 − 0.0329i)17-s + (−0.473 − 0.819i)19-s + (−0.0770 + 0.490i)20-s + (0.539 − 0.539i)22-s + (−0.139 + 0.520i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.543 + 0.839i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.543 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40420 - 0.763223i\)
\(L(\frac12)\) \(\approx\) \(1.40420 - 0.763223i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.803 + 2.08i)T \)
7 \( 1 + (-2.42 - 1.06i)T \)
good2 \( 1 + (-0.969 - 0.259i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-1.78 + 3.08i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.78 + 2.78i)T - 13iT^{2} \)
17 \( 1 + (-0.506 + 0.135i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.06 + 3.57i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.668 - 2.49i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 6.14iT - 29T^{2} \)
31 \( 1 + (1.71 + 0.988i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.152 - 0.0409i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 8.28iT - 41T^{2} \)
43 \( 1 + (-9.01 - 9.01i)T + 43iT^{2} \)
47 \( 1 + (-1.39 + 5.19i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.55 - 1.48i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.30 - 2.25i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.67 + 5.00i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.42 + 5.32i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.23T + 71T^{2} \)
73 \( 1 + (3.98 + 14.8i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (13.1 - 7.57i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.42 + 9.42i)T - 83iT^{2} \)
89 \( 1 + (-5.52 - 9.57i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.48 - 2.48i)T + 97iT^{2} \)
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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

−11.61212591038098204107715197968, −10.85491803182861651779373123899, −9.324162603040010981593016420011, −8.721062123103668055676990272019, −7.88615086005352391643543390554, −6.22082876407273794242968844205, −5.37533324668169849461031495261, −4.54877252540457532599701011568, −3.42398353279116780162075104986, −1.07766039244548292990330050736, 2.17711490314391615922524413074, 3.89922261345265255045272541563, 4.27812608106800919462498971371, 5.77902117603600150856496259502, 6.96392962723211385415758568055, 7.951435275540006672099065555898, 8.893916080294666903257046957745, 10.14539266488425514665260359931, 11.12240798099713340406413263857, 11.84140627559776678244149426148

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