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Properties

Label 2-315-35.2-c2-0-35
Degree $2$
Conductor $315$
Sign $0.328 + 0.944i$
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  + (3.56 − 0.955i)2-s + (8.33 − 4.81i)4-s + (1.05 − 4.88i)5-s + (−6.28 + 3.07i)7-s + (14.6 − 14.6i)8-s + (−0.890 − 18.4i)10-s + (4.09 + 7.09i)11-s + (14.0 − 14.0i)13-s + (−19.4 + 16.9i)14-s + (19.0 − 32.9i)16-s + (−1.81 + 6.76i)17-s + (−18.2 − 10.5i)19-s + (−14.6 − 45.8i)20-s + (21.3 + 21.3i)22-s + (8.43 + 31.4i)23-s + ⋯
L(s)  = 1  + (1.78 − 0.477i)2-s + (2.08 − 1.20i)4-s + (0.211 − 0.977i)5-s + (−0.898 + 0.439i)7-s + (1.83 − 1.83i)8-s + (−0.0890 − 1.84i)10-s + (0.372 + 0.644i)11-s + (1.08 − 1.08i)13-s + (−1.39 + 1.21i)14-s + (1.18 − 2.06i)16-s + (−0.106 + 0.397i)17-s + (−0.959 − 0.554i)19-s + (−0.733 − 2.29i)20-s + (0.971 + 0.971i)22-s + (0.366 + 1.36i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.328 + 0.944i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ 0.328 + 0.944i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.67079 - 2.61025i\)
\(L(\frac12)\) \(\approx\) \(3.67079 - 2.61025i\)
\(L(2)\) 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 \( 1 + (-1.05 + 4.88i)T \)
7 \( 1 + (6.28 - 3.07i)T \)
good2 \( 1 + (-3.56 + 0.955i)T + (3.46 - 2i)T^{2} \)
11 \( 1 + (-4.09 - 7.09i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-14.0 + 14.0i)T - 169iT^{2} \)
17 \( 1 + (1.81 - 6.76i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (18.2 + 10.5i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-8.43 - 31.4i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 - 22.1iT - 841T^{2} \)
31 \( 1 + (-13.7 - 23.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-14.9 + 4.01i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 0.496T + 1.68e3T^{2} \)
43 \( 1 + (33.4 - 33.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (24.1 - 6.46i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-34.3 - 9.20i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-15.5 + 9.00i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-13.4 + 23.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (1.44 - 5.37i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 105.T + 5.04e3T^{2} \)
73 \( 1 + (81.6 + 21.8i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (86.5 + 49.9i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (95.6 - 95.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (54.3 + 31.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-59.0 - 59.0i)T + 9.40e3iT^{2} \)
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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

−11.63037271627317509680201861378, −10.63980099306125688351781186354, −9.627320102625989970775419840831, −8.472693717188537583512446910266, −6.82147522953042285380896558680, −5.93606467307572674362432594032, −5.15911612608182749651214125315, −4.04193188760450265677365975121, −3.01896567727575963589802782225, −1.51113908833122195722256917390, 2.44594451073303117293777761297, 3.59243016260259349323424865255, 4.27999962673391520965542864933, 6.00235298013819852958449941208, 6.41869056245117773175263734865, 7.11118617623398431401021327976, 8.540570228159185938069536590832, 10.08374088288161328017614042028, 11.10754588611079793805727457792, 11.71356060308754558186972129264

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