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Properties

Label 2-315-315.313-c1-0-43
Degree $2$
Conductor $315$
Sign $-0.428 - 0.903i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.133i)2-s − 1.73i·3-s + (−1.5 − 0.866i)4-s + (−2 − i)5-s + (−0.232 + 0.866i)6-s + (−0.866 + 2.5i)7-s + (1.36 + 1.36i)8-s − 2.99·9-s + (0.866 + 0.767i)10-s + 1.46·11-s + (−1.49 + 2.59i)12-s + (−1 − 0.267i)13-s + (0.767 − 1.13i)14-s + (−1.73 + 3.46i)15-s + (1.23 + 2.13i)16-s + (0.732 − 2.73i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.0947i)2-s − 0.999i·3-s + (−0.750 − 0.433i)4-s + (−0.894 − 0.447i)5-s + (−0.0947 + 0.353i)6-s + (−0.327 + 0.944i)7-s + (0.482 + 0.482i)8-s − 0.999·9-s + (0.273 + 0.242i)10-s + 0.441·11-s + (−0.433 + 0.749i)12-s + (−0.277 − 0.0743i)13-s + (0.205 − 0.303i)14-s + (−0.447 + 0.894i)15-s + (0.308 + 0.533i)16-s + (0.177 − 0.662i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.428 - 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.428 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.428 - 0.903i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.428 - 0.903i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
5 \( 1 + (2 + i)T \)
7 \( 1 + (0.866 - 2.5i)T \)
good2 \( 1 + (0.5 + 0.133i)T + (1.73 + i)T^{2} \)
11 \( 1 - 1.46T + 11T^{2} \)
13 \( 1 + (1 + 0.267i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.732 + 2.73i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.36 - 5.83i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.63 - 1.63i)T + 23iT^{2} \)
29 \( 1 + (6 + 3.46i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (8.83 + 5.09i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.09 - 4.09i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (8.19 - 4.73i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.59 - 1.5i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-2.30 + 8.59i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.901 + 3.36i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.73 + 3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.13 - 4.69i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.23 - 8.33i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (0.633 + 0.169i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-9.92 + 5.73i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.366 + 1.36i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-0.598 + 1.03i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.09 + 0.830i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.33467563382244215040251815033, −9.866114136006841438785357161225, −8.948397352546535760158222975512, −8.294441096958454315038611985753, −7.39482373608022047951746382514, −6.02458226459528344070370604633, −5.12741577401403385966039937752, −3.61559866819182776119365940713, −1.77617134962978921523282324780, 0, 3.36016540368344228656006488177, 4.01157957633073386937850796672, 4.94481727418217753262131829935, 6.75050716075505101831522457435, 7.61532154799655439367352683168, 8.741927061907267217657628629758, 9.346650911331807747779136949673, 10.59210408199233913095640632076, 10.91413895052672891492728937430

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