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Properties

Label 2-315-7.3-c2-0-22
Degree $2$
Conductor $315$
Sign $-0.0974 + 0.995i$
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  + (0.336 + 0.583i)2-s + (1.77 − 3.07i)4-s + (−1.93 + 1.11i)5-s + (−6.82 − 1.55i)7-s + 5.08·8-s + (−1.30 − 0.752i)10-s + (−0.0223 + 0.0387i)11-s − 23.0i·13-s + (−1.38 − 4.50i)14-s + (−5.38 − 9.32i)16-s + (8.16 + 4.71i)17-s + (0.991 − 0.572i)19-s + 7.93i·20-s − 0.0301·22-s + (−22.1 − 38.3i)23-s + ⋯
L(s)  = 1  + (0.168 + 0.291i)2-s + (0.443 − 0.767i)4-s + (−0.387 + 0.223i)5-s + (−0.974 − 0.222i)7-s + 0.635·8-s + (−0.130 − 0.0752i)10-s + (−0.00203 + 0.00352i)11-s − 1.76i·13-s + (−0.0992 − 0.321i)14-s + (−0.336 − 0.582i)16-s + (0.480 + 0.277i)17-s + (0.0521 − 0.0301i)19-s + 0.396i·20-s − 0.00137·22-s + (−0.961 − 1.66i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0974 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0974 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0974 + 0.995i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ -0.0974 + 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.877306 - 0.967415i\)
\(L(\frac12)\) \(\approx\) \(0.877306 - 0.967415i\)
\(L(2)\) 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 \( 1 + (6.82 + 1.55i)T \)
good2 \( 1 + (-0.336 - 0.583i)T + (-2 + 3.46i)T^{2} \)
11 \( 1 + (0.0223 - 0.0387i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 23.0iT - 169T^{2} \)
17 \( 1 + (-8.16 - 4.71i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-0.991 + 0.572i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (22.1 + 38.3i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 53.0T + 841T^{2} \)
31 \( 1 + (-19.5 - 11.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (21.1 + 36.6i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 38.2iT - 1.68e3T^{2} \)
43 \( 1 - 76.5T + 1.84e3T^{2} \)
47 \( 1 + (-23.5 + 13.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-9.49 + 16.4i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-4.21 - 2.43i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (33.6 - 19.4i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-3.50 + 6.06i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 46.8T + 5.04e3T^{2} \)
73 \( 1 + (-72.3 - 41.7i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (10.2 + 17.7i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 125. iT - 6.88e3T^{2} \)
89 \( 1 + (40.4 - 23.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 3.11iT - 9.40e3T^{2} \)
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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

−10.86780829673897436084522747514, −10.42356832264941685474646159707, −9.547652261439355222696484624276, −8.105480201104380444344183555289, −7.23329827529742786859611212896, −6.19693495105786354606341761860, −5.45748647870396266554201944208, −3.92387936273947543136933607259, −2.61930884512760112830373891300, −0.56779936941290883257075542879, 1.97773098533564910692859584198, 3.42523150853770625078864138850, 4.20379691378134948666829796084, 5.83823385043584015538769722174, 7.00274278018513507596927194056, 7.70704586227643186353207600732, 9.009730883964101703063285313103, 9.694878722367706250039357860874, 11.06612176956912307487765712614, 11.85404050621241572937512779429

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