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Properties

Label 2-315-63.41-c1-0-30
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: $\chi_{315} (41, \cdot )$
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

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

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