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Properties

Label 2-315-63.25-c1-0-13
Degree $2$
Conductor $315$
Sign $0.500 + 0.865i$
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  − 0.259·2-s + (−1.21 + 1.23i)3-s − 1.93·4-s + (−0.5 + 0.866i)5-s + (0.316 − 0.319i)6-s + (−0.593 − 2.57i)7-s + 1.02·8-s + (−0.0368 − 2.99i)9-s + (0.129 − 0.224i)10-s + (0.726 + 1.25i)11-s + (2.35 − 2.38i)12-s + (−0.192 − 0.333i)13-s + (0.154 + 0.669i)14-s + (−0.458 − 1.67i)15-s + 3.59·16-s + (2.42 − 4.20i)17-s + ⋯
L(s)  = 1  − 0.183·2-s + (−0.702 + 0.711i)3-s − 0.966·4-s + (−0.223 + 0.387i)5-s + (0.129 − 0.130i)6-s + (−0.224 − 0.974i)7-s + 0.361·8-s + (−0.0122 − 0.999i)9-s + (0.0410 − 0.0711i)10-s + (0.219 + 0.379i)11-s + (0.679 − 0.687i)12-s + (−0.0533 − 0.0923i)13-s + (0.0411 + 0.178i)14-s + (−0.118 − 0.431i)15-s + 0.899·16-s + (0.588 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.500 + 0.865i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.500 + 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.461097 - 0.266031i\)
\(L(\frac12)\) \(\approx\) \(0.461097 - 0.266031i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.21 - 1.23i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.593 + 2.57i)T \)
good2 \( 1 + 0.259T + 2T^{2} \)
11 \( 1 + (-0.726 - 1.25i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.192 + 0.333i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.42 + 4.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.194 - 0.337i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.97 + 6.88i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.48 + 4.30i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.39T + 31T^{2} \)
37 \( 1 + (5.98 + 10.3i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.47 - 6.02i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.06 - 1.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.62T + 47T^{2} \)
53 \( 1 + (-2.14 + 3.71i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.82T + 59T^{2} \)
61 \( 1 - 7.75T + 61T^{2} \)
67 \( 1 + 4.30T + 67T^{2} \)
71 \( 1 + 1.38T + 71T^{2} \)
73 \( 1 + (-5.49 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 1.67T + 79T^{2} \)
83 \( 1 + (-6.11 + 10.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.23 + 5.61i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.759 - 1.31i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.27725153988415647612758164119, −10.44309739809149788014192223190, −9.789976200819743781691489988558, −8.963202689638375737812382401567, −7.61400172441199235704324760030, −6.66741781816184288344449762275, −5.27407280928243688252453666820, −4.37944961681600160330862881739, −3.44926209937716101394081596396, −0.50790075217311744231136525861, 1.43544051926428060911367036263, 3.53965173282448560842878891637, 5.14379462921029878000264262528, 5.65433246014363224220664616234, 7.00485957177762671710115058146, 8.198577036604089099562868830186, 8.836206290204065283865623814007, 9.864328643002827304636504847890, 11.02324452257414530441022842103, 11.97787583026807066615029777178

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