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Properties

Label 2-315-7.4-c3-0-21
Degree $2$
Conductor $315$
Sign $0.723 + 0.690i$
Analytic cond. $18.5856$
Root an. cond. $4.31110$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.720 − 1.24i)2-s + (2.96 − 5.13i)4-s + (2.5 + 4.33i)5-s + (18.2 + 3.22i)7-s − 20.0·8-s + (3.60 − 6.23i)10-s + (−5.12 + 8.87i)11-s + 64.1·13-s + (−9.11 − 25.0i)14-s + (−9.25 − 16.0i)16-s + (10.9 − 18.8i)17-s + (55.4 + 96.1i)19-s + 29.6·20-s + 14.7·22-s + (29.9 + 51.8i)23-s + ⋯
L(s)  = 1  + (−0.254 − 0.441i)2-s + (0.370 − 0.641i)4-s + (0.223 + 0.387i)5-s + (0.984 + 0.174i)7-s − 0.886·8-s + (0.113 − 0.197i)10-s + (−0.140 + 0.243i)11-s + 1.36·13-s + (−0.173 − 0.478i)14-s + (−0.144 − 0.250i)16-s + (0.155 − 0.269i)17-s + (0.670 + 1.16i)19-s + 0.331·20-s + 0.143·22-s + (0.271 + 0.470i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 + 0.690i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.723 + 0.690i$
Analytic conductor: \(18.5856\)
Root analytic conductor: \(4.31110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :3/2),\ 0.723 + 0.690i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.147719713\)
\(L(\frac12)\) \(\approx\) \(2.147719713\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.5 - 4.33i)T \)
7 \( 1 + (-18.2 - 3.22i)T \)
good2 \( 1 + (0.720 + 1.24i)T + (-4 + 6.92i)T^{2} \)
11 \( 1 + (5.12 - 8.87i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 64.1T + 2.19e3T^{2} \)
17 \( 1 + (-10.9 + 18.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-55.4 - 96.1i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-29.9 - 51.8i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 22.2T + 2.43e4T^{2} \)
31 \( 1 + (-34.9 + 60.4i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (176. + 305. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 378.T + 6.89e4T^{2} \)
43 \( 1 - 30.1T + 7.95e4T^{2} \)
47 \( 1 + (20.7 + 35.9i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-252. + 437. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-269. + 466. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (176. + 305. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-116. + 200. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 723.T + 3.57e5T^{2} \)
73 \( 1 + (529. - 917. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-83.4 - 144. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 924.T + 5.71e5T^{2} \)
89 \( 1 + (496. + 859. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 744.T + 9.12e5T^{2} \)
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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

−11.13106533737730316009423346351, −10.32400375547037583762688443785, −9.422826659486779019818717007711, −8.380582442317214275061752229335, −7.28403656645745840426146159560, −6.03712766024929907060919148688, −5.31386904898557916011420886407, −3.65565385596705709332454895149, −2.18523721576198602318542350191, −1.13648663563468330559242619539, 1.16125601582875576190518848833, 2.83082883737908951855221857088, 4.21561100997831392673047587461, 5.49451570324665608783412067722, 6.58027969511954511573682831910, 7.62401951147319552091452851653, 8.491224079194700445354153697507, 9.036417345528688412592510009269, 10.58157506024559806494580267775, 11.34403478066675768521314643329

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