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Properties

Label 2-315-63.20-c1-0-13
Degree $2$
Conductor $315$
Sign $-0.459 - 0.888i$
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  + (2.31 + 1.33i)2-s + (−0.852 + 1.50i)3-s + (2.58 + 4.48i)4-s + (−0.5 − 0.866i)5-s + (−3.99 + 2.35i)6-s + (2.62 − 0.288i)7-s + 8.50i·8-s + (−1.54 − 2.56i)9-s − 2.67i·10-s + (−4.13 − 2.38i)11-s + (−8.96 + 0.0835i)12-s + (3.81 − 2.20i)13-s + (6.48 + 2.85i)14-s + (1.73 − 0.0161i)15-s + (−6.22 + 10.7i)16-s − 2.88·17-s + ⋯
L(s)  = 1  + (1.64 + 0.947i)2-s + (−0.491 + 0.870i)3-s + (1.29 + 2.24i)4-s + (−0.223 − 0.387i)5-s + (−1.63 + 0.962i)6-s + (0.994 − 0.109i)7-s + 3.00i·8-s + (−0.516 − 0.856i)9-s − 0.847i·10-s + (−1.24 − 0.719i)11-s + (−2.58 + 0.0241i)12-s + (1.05 − 0.610i)13-s + (1.73 + 0.762i)14-s + (0.447 − 0.00416i)15-s + (−1.55 + 2.69i)16-s − 0.699·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.459 - 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.459 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.459 - 0.888i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.459 - 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41102 + 2.31743i\)
\(L(\frac12)\) \(\approx\) \(1.41102 + 2.31743i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.852 - 1.50i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.62 + 0.288i)T \)
good2 \( 1 + (-2.31 - 1.33i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (4.13 + 2.38i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.81 + 2.20i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.88T + 17T^{2} \)
19 \( 1 + 1.11iT - 19T^{2} \)
23 \( 1 + (-0.0967 + 0.0558i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.32 - 3.65i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.39 - 1.96i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.197T + 37T^{2} \)
41 \( 1 + (4.21 + 7.30i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.01 - 1.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.67 - 6.35i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 10.3iT - 53T^{2} \)
59 \( 1 + (-2.59 - 4.49i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.71 - 2.14i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.20 + 5.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.343iT - 71T^{2} \)
73 \( 1 + 7.87iT - 73T^{2} \)
79 \( 1 + (6.55 - 11.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.77 - 6.53i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.94T + 89T^{2} \)
97 \( 1 + (14.8 + 8.59i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.11015758676908913628732004335, −11.15222931260986798607910691429, −10.69304869409237514004109725557, −8.615537487941348459470217094898, −8.105558445798079718075000951619, −6.75355746519430042047091701127, −5.56947438557542863595185854790, −5.10797917332499581369860871871, −4.15267791444101445025947075620, −3.06220773461058713664718986779, 1.66463204475235561441766087176, 2.64355512929892572636668953232, 4.28100240847635848187427989043, 5.14824368160523759663592723764, 6.13027799043356275972420731580, 7.08809279224541042336162037666, 8.276127272940651330931121525574, 10.20426460536911032631367210451, 10.97947514349030005800235127924, 11.52578918432372663015665963087

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