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Properties

Label 2-315-7.5-c2-0-22
Degree $2$
Conductor $315$
Sign $-0.577 + 0.816i$
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  + (1.26 − 2.18i)2-s + (−1.17 − 2.03i)4-s + (1.93 + 1.11i)5-s + (−6.18 − 3.28i)7-s + 4.15·8-s + (4.88 − 2.81i)10-s + (−4.36 − 7.55i)11-s − 21.5i·13-s + (−14.9 + 9.34i)14-s + (9.93 − 17.2i)16-s + (18.7 − 10.8i)17-s + (−2.71 − 1.56i)19-s − 5.26i·20-s − 21.9·22-s + (2.05 − 3.55i)23-s + ⋯
L(s)  = 1  + (0.630 − 1.09i)2-s + (−0.294 − 0.509i)4-s + (0.387 + 0.223i)5-s + (−0.882 − 0.469i)7-s + 0.519·8-s + (0.488 − 0.281i)10-s + (−0.396 − 0.686i)11-s − 1.65i·13-s + (−1.06 + 0.667i)14-s + (0.621 − 1.07i)16-s + (1.10 − 0.638i)17-s + (−0.142 − 0.0825i)19-s − 0.263i·20-s − 0.999·22-s + (0.0893 − 0.154i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ -0.577 + 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.03668 - 2.00272i\)
\(L(\frac12)\) \(\approx\) \(1.03668 - 2.00272i\)
\(L(2)\) 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 \( 1 + (-1.93 - 1.11i)T \)
7 \( 1 + (6.18 + 3.28i)T \)
good2 \( 1 + (-1.26 + 2.18i)T + (-2 - 3.46i)T^{2} \)
11 \( 1 + (4.36 + 7.55i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 21.5iT - 169T^{2} \)
17 \( 1 + (-18.7 + 10.8i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (2.71 + 1.56i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-2.05 + 3.55i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 50.8T + 841T^{2} \)
31 \( 1 + (33.9 - 19.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (26.4 - 45.8i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 36.8iT - 1.68e3T^{2} \)
43 \( 1 - 17.6T + 1.84e3T^{2} \)
47 \( 1 + (-3.49 - 2.01i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-2.22 - 3.85i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (81.5 - 47.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (63.3 + 36.5i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-50.2 - 87.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 56.6T + 5.04e3T^{2} \)
73 \( 1 + (-64.8 + 37.4i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (14.4 - 25.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 21.1iT - 6.88e3T^{2} \)
89 \( 1 + (63.1 + 36.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 73.7iT - 9.40e3T^{2} \)
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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

−11.01411441016767456970906007130, −10.36003377296261460538014131727, −9.806817220791712221645652145168, −8.265265853782404083470617118186, −7.22706822264289797796473863013, −5.91353898346077917437824718285, −4.88644426748013477318801034446, −3.25876799667005797001834453104, −2.94873902723945463057061233873, −0.925612409576593783758972072363, 1.97411318080235680114561315113, 3.83040280955542916285942229374, 4.98460999474399611264616902403, 5.95551824137194042587792762847, 6.69903832529037950702334741491, 7.62113583149271664045988464190, 8.885021368474633467951357090990, 9.756802265693317249709818735416, 10.71653889202140929601254195145, 12.23195795568762890992161854706

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