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Properties

Label 2-315-5.2-c2-0-29
Degree $2$
Conductor $315$
Sign $-0.492 - 0.870i$
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  + (−0.675 − 0.675i)2-s − 3.08i·4-s + (−3.39 − 3.67i)5-s + (−1.87 − 1.87i)7-s + (−4.78 + 4.78i)8-s + (−0.186 + 4.77i)10-s − 7.59·11-s + (1.12 − 1.12i)13-s + 2.52i·14-s − 5.88·16-s + (3.43 + 3.43i)17-s + 26.3i·19-s + (−11.3 + 10.4i)20-s + (5.13 + 5.13i)22-s + (24.2 − 24.2i)23-s + ⋯
L(s)  = 1  + (−0.337 − 0.337i)2-s − 0.771i·4-s + (−0.678 − 0.734i)5-s + (−0.267 − 0.267i)7-s + (−0.598 + 0.598i)8-s + (−0.0186 + 0.477i)10-s − 0.690·11-s + (0.0863 − 0.0863i)13-s + 0.180i·14-s − 0.367·16-s + (0.202 + 0.202i)17-s + 1.38i·19-s + (−0.566 + 0.524i)20-s + (0.233 + 0.233i)22-s + (1.05 − 1.05i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.492 - 0.870i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ -0.492 - 0.870i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0769401 + 0.131881i\)
\(L(\frac12)\) \(\approx\) \(0.0769401 + 0.131881i\)
\(L(2)\) 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 \( 1 + (3.39 + 3.67i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good2 \( 1 + (0.675 + 0.675i)T + 4iT^{2} \)
11 \( 1 + 7.59T + 121T^{2} \)
13 \( 1 + (-1.12 + 1.12i)T - 169iT^{2} \)
17 \( 1 + (-3.43 - 3.43i)T + 289iT^{2} \)
19 \( 1 - 26.3iT - 361T^{2} \)
23 \( 1 + (-24.2 + 24.2i)T - 529iT^{2} \)
29 \( 1 - 22.3iT - 841T^{2} \)
31 \( 1 + 18.3T + 961T^{2} \)
37 \( 1 + (34.4 + 34.4i)T + 1.36e3iT^{2} \)
41 \( 1 + 37.4T + 1.68e3T^{2} \)
43 \( 1 + (55.1 - 55.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (40.8 + 40.8i)T + 2.20e3iT^{2} \)
53 \( 1 + (-9.39 + 9.39i)T - 2.80e3iT^{2} \)
59 \( 1 - 49.7iT - 3.48e3T^{2} \)
61 \( 1 - 88.3T + 3.72e3T^{2} \)
67 \( 1 + (40.5 + 40.5i)T + 4.48e3iT^{2} \)
71 \( 1 + 136.T + 5.04e3T^{2} \)
73 \( 1 + (-5.85 + 5.85i)T - 5.32e3iT^{2} \)
79 \( 1 + 66.4iT - 6.24e3T^{2} \)
83 \( 1 + (-34.8 + 34.8i)T - 6.88e3iT^{2} \)
89 \( 1 + 2.03iT - 7.92e3T^{2} \)
97 \( 1 + (58.4 + 58.4i)T + 9.40e3iT^{2} \)
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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

−10.65882416240579740738991317409, −10.06235023179237958122860092624, −8.912736840739167228042836271197, −8.214959444392681247571151217425, −6.98323818843873710434703288392, −5.66668927802540926560765022047, −4.78399622260089656876388103953, −3.33892477521755348001644888127, −1.54469021593510614400382713887, −0.081284701308068643006915504704, 2.74994163814172381967209673455, 3.59560559857062104867896558983, 5.08262795997699035178811931722, 6.66117189574502967496396546930, 7.25263809990532572275122839913, 8.185829429140628045726538311418, 9.061387965716923247758408434426, 10.14461872690001237287270917575, 11.32501269010519530977173700268, 11.84590365187679639483467322732

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