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Properties

Label 2-315-315.283-c1-0-33
Degree $2$
Conductor $315$
Sign $0.999 - 0.0212i$
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  + (0.133 + 0.5i)2-s + (1.5 + 0.866i)3-s + (1.5 − 0.866i)4-s + (−1 − 2i)5-s + (−0.232 + 0.866i)6-s + (0.5 − 2.59i)7-s + (1.36 + 1.36i)8-s + (1.5 + 2.59i)9-s + (0.866 − 0.767i)10-s − 2·11-s + 3·12-s + (−0.0980 − 0.366i)13-s + (1.36 − 0.0980i)14-s + (0.232 − 3.86i)15-s + (1.23 − 2.13i)16-s + (−2 + 0.535i)17-s + ⋯
L(s)  = 1  + (0.0947 + 0.353i)2-s + (0.866 + 0.499i)3-s + (0.750 − 0.433i)4-s + (−0.447 − 0.894i)5-s + (−0.0947 + 0.353i)6-s + (0.188 − 0.981i)7-s + (0.482 + 0.482i)8-s + (0.5 + 0.866i)9-s + (0.273 − 0.242i)10-s − 0.603·11-s + 0.866·12-s + (−0.0272 − 0.101i)13-s + (0.365 − 0.0262i)14-s + (0.0599 − 0.998i)15-s + (0.308 − 0.533i)16-s + (−0.485 + 0.129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0212i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.999 - 0.0212i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.999 - 0.0212i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97389 + 0.0209546i\)
\(L(\frac12)\) \(\approx\) \(1.97389 + 0.0209546i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 - 0.866i)T \)
5 \( 1 + (1 + 2i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
good2 \( 1 + (-0.133 - 0.5i)T + (-1.73 + i)T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (0.0980 + 0.366i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (2 - 0.535i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.366 - 0.633i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.46 - 4.46i)T + 23iT^{2} \)
29 \( 1 + (8.59 - 4.96i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.73 - 2.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.83 - 2.36i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.59 + 1.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.232 - 0.866i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-4.96 + 1.33i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1 - 0.267i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (4.56 + 7.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.46 - 5.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.36 + 0.901i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 1.26T + 71T^{2} \)
73 \( 1 + (3.46 + 12.9i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (9.92 + 5.73i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.96 + 2.66i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (0.535 + 0.928i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.43 + 9.09i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.35852402576573891375359517005, −10.77229746565279475065815360607, −9.762627440641598398655608568771, −8.768748725277002489205612415799, −7.66147391035983883436754145444, −7.24670061563570118686494457134, −5.48586747237918692490551216818, −4.60945509009447483776951523236, −3.37796448388086719824538128949, −1.63954039310203090510116205448, 2.24468932413022242672398685882, 2.82203297716958313937660317349, 4.06187117146737546223648756903, 6.00551476480828348802929392082, 7.07684825093814966878338681340, 7.72484850768240145676932025589, 8.671906318967634134299020945612, 9.804686164020875589959890183240, 11.07556100262351248766184522042, 11.54498396271546772508030628328

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