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Properties

Label 2-315-63.2-c2-0-30
Degree $2$
Conductor $315$
Sign $0.904 + 0.426i$
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  + (−3.13 + 1.80i)2-s + (−2.12 − 2.11i)3-s + (4.53 − 7.85i)4-s − 2.23i·5-s + (10.4 + 2.79i)6-s + (6.87 − 1.32i)7-s + 18.3i·8-s + (0.0264 + 8.99i)9-s + (4.04 + 7.00i)10-s − 10.6i·11-s + (−26.2 + 7.08i)12-s + (10.3 + 17.9i)13-s + (−19.1 + 16.5i)14-s + (−4.73 + 4.75i)15-s + (−15.0 − 26.0i)16-s + (−12.3 + 7.14i)17-s + ⋯
L(s)  = 1  + (−1.56 + 0.903i)2-s + (−0.708 − 0.706i)3-s + (1.13 − 1.96i)4-s − 0.447i·5-s + (1.74 + 0.465i)6-s + (0.981 − 0.189i)7-s + 2.29i·8-s + (0.00293 + 0.999i)9-s + (0.404 + 0.700i)10-s − 0.965i·11-s + (−2.19 + 0.590i)12-s + (0.795 + 1.37i)13-s + (−1.36 + 1.18i)14-s + (−0.315 + 0.316i)15-s + (−0.938 − 1.62i)16-s + (−0.727 + 0.420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.904 + 0.426i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ 0.904 + 0.426i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.635719 - 0.142542i\)
\(L(\frac12)\) \(\approx\) \(0.635719 - 0.142542i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.12 + 2.11i)T \)
5 \( 1 + 2.23iT \)
7 \( 1 + (-6.87 + 1.32i)T \)
good2 \( 1 + (3.13 - 1.80i)T + (2 - 3.46i)T^{2} \)
11 \( 1 + 10.6iT - 121T^{2} \)
13 \( 1 + (-10.3 - 17.9i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (12.3 - 7.14i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-14.2 + 24.6i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 - 14.7iT - 529T^{2} \)
29 \( 1 + (-29.3 - 16.9i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-5.14 + 8.90i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-3.55 + 6.15i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (3.48 - 2.01i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (23.5 - 40.7i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-62.2 + 35.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-38.7 + 22.3i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-40.2 - 23.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (4.34 + 7.53i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-12.4 + 21.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 134. iT - 5.04e3T^{2} \)
73 \( 1 + (68.0 + 117. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (36.7 + 63.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (47.7 + 27.5i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (8.61 + 4.97i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-12.7 + 22.0i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.28963316205763897417069200654, −10.47075777391661516604679842445, −9.005048720657300560452017809523, −8.579377012887416962176594697925, −7.57572883819867089835150028678, −6.74100412075324400671688152017, −5.89137971338782719189351559499, −4.75838083160319635681779762205, −1.77075300506521363652637269427, −0.76461850168558373348329909002, 1.05539426606001396885916479640, 2.64937604324051162151145298679, 4.06219009821867044326330291312, 5.54910246559034174374513996831, 7.00127672655255176816131663328, 8.058464218893830219819118071941, 8.833878255933600149007481084138, 10.14790690936647641754061405002, 10.27665084515402277007514915097, 11.26306098792476878608259489770

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