Properties

Label 2-315-35.17-c1-0-7
Degree $2$
Conductor $315$
Sign $0.607 - 0.794i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.602 + 2.24i)2-s + (−2.95 − 1.70i)4-s + (1.28 − 1.82i)5-s + (0.519 − 2.59i)7-s + (2.33 − 2.33i)8-s + (3.33 + 3.99i)10-s + (1.76 − 3.05i)11-s + (4.49 + 4.49i)13-s + (5.51 + 2.73i)14-s + (0.421 + 0.729i)16-s + (−0.481 − 1.79i)17-s + (−0.0699 − 0.121i)19-s + (−6.92 + 3.21i)20-s + (5.80 + 5.80i)22-s + (3.72 + 0.997i)23-s + ⋯
L(s)  = 1  + (−0.425 + 1.58i)2-s + (−1.47 − 0.854i)4-s + (0.574 − 0.818i)5-s + (0.196 − 0.980i)7-s + (0.824 − 0.824i)8-s + (1.05 + 1.26i)10-s + (0.531 − 0.921i)11-s + (1.24 + 1.24i)13-s + (1.47 + 0.730i)14-s + (0.105 + 0.182i)16-s + (−0.116 − 0.436i)17-s + (−0.0160 − 0.0277i)19-s + (−1.54 + 0.719i)20-s + (1.23 + 1.23i)22-s + (0.775 + 0.207i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.607 - 0.794i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.607 - 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01811 + 0.502752i\)
\(L(\frac12)\) \(\approx\) \(1.01811 + 0.502752i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-1.28 + 1.82i)T \)
7 \( 1 + (-0.519 + 2.59i)T \)
good2 \( 1 + (0.602 - 2.24i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-1.76 + 3.05i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.49 - 4.49i)T + 13iT^{2} \)
17 \( 1 + (0.481 + 1.79i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.0699 + 0.121i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.72 - 0.997i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 2.01iT - 29T^{2} \)
31 \( 1 + (4.56 + 2.63i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.50 - 5.61i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.903iT - 41T^{2} \)
43 \( 1 + (-2.38 + 2.38i)T - 43iT^{2} \)
47 \( 1 + (-2.38 - 0.639i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.726 - 2.71i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.15 - 5.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.69 - 5.01i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.3 + 2.77i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 5.09T + 71T^{2} \)
73 \( 1 + (-9.04 + 2.42i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-7.30 + 4.21i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.37 + 7.37i)T + 83iT^{2} \)
89 \( 1 + (1.75 + 3.03i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.70 - 8.70i)T - 97iT^{2} \)
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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

−11.69322859440735228423295887927, −10.70694630320728138658116610462, −9.277894083861601259388183023361, −8.957206177766151902411273324980, −7.978475988687699081145736915805, −6.85866803380922977392685964735, −6.16245505695604831227769435230, −5.09021072736365848363437400656, −3.97768200590430555214332833574, −1.13537372894706861699433088160, 1.63935224072062048782724734445, 2.72889578878262894963381654862, 3.77463743096397028480316805605, 5.43850107818581672647378326155, 6.59124490511141186210248467850, 8.137328761155501401770985269584, 9.108191008925198752193945170518, 9.800207654367391580500790325413, 10.85100453064198621614047703104, 11.14527570783461435526250718026

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