Properties

Label 2-315-35.2-c2-0-1
Degree $2$
Conductor $315$
Sign $-0.998 + 0.0566i$
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  + (−0.984 + 0.263i)2-s + (−2.56 + 1.48i)4-s + (4.04 + 2.93i)5-s + (−5.81 + 3.89i)7-s + (5.01 − 5.01i)8-s + (−4.75 − 1.81i)10-s + (2.58 + 4.48i)11-s + (−3.12 + 3.12i)13-s + (4.69 − 5.36i)14-s + (2.30 − 3.99i)16-s + (−4.02 + 15.0i)17-s + (−17.1 − 9.92i)19-s + (−14.7 − 1.52i)20-s + (−3.73 − 3.73i)22-s + (−9.71 − 36.2i)23-s + ⋯
L(s)  = 1  + (−0.492 + 0.131i)2-s + (−0.641 + 0.370i)4-s + (0.809 + 0.586i)5-s + (−0.830 + 0.556i)7-s + (0.627 − 0.627i)8-s + (−0.475 − 0.181i)10-s + (0.235 + 0.407i)11-s + (−0.240 + 0.240i)13-s + (0.335 − 0.383i)14-s + (0.144 − 0.249i)16-s + (−0.236 + 0.884i)17-s + (−0.904 − 0.522i)19-s + (−0.736 − 0.0762i)20-s + (−0.169 − 0.169i)22-s + (−0.422 − 1.57i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0566i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.998 + 0.0566i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ -0.998 + 0.0566i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0124439 - 0.439142i\)
\(L(\frac12)\) \(\approx\) \(0.0124439 - 0.439142i\)
\(L(2)\) 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 \( 1 + (-4.04 - 2.93i)T \)
7 \( 1 + (5.81 - 3.89i)T \)
good2 \( 1 + (0.984 - 0.263i)T + (3.46 - 2i)T^{2} \)
11 \( 1 + (-2.58 - 4.48i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (3.12 - 3.12i)T - 169iT^{2} \)
17 \( 1 + (4.02 - 15.0i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (17.1 + 9.92i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (9.71 + 36.2i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 - 11.8iT - 841T^{2} \)
31 \( 1 + (9.02 + 15.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (70.9 - 19.0i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 60.6T + 1.68e3T^{2} \)
43 \( 1 + (33.1 - 33.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (-19.9 + 5.33i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-6.28 - 1.68i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-32.2 + 18.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (6.88 - 11.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (0.969 - 3.61i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 77.9T + 5.04e3T^{2} \)
73 \( 1 + (123. + 33.1i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-73.5 - 42.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-10.0 + 10.0i)T - 6.88e3iT^{2} \)
89 \( 1 + (-95.4 - 55.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-30.0 - 30.0i)T + 9.40e3iT^{2} \)
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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

−12.09408035954760871059657570750, −10.54472646037055071462791806631, −10.02251094659983777853057206891, −9.054099846637036172041694465177, −8.436002896638382378975580632121, −6.93039706362812557302192163769, −6.35269214450313266086266592339, −4.90474097943407660233140147852, −3.56044653290260306237152379569, −2.12908356507807373568884056890, 0.24333177877106517398727700292, 1.74566017364646265174849272188, 3.65180571670715462419093372327, 5.00624540541173329563934584131, 5.88849248871790678437963208258, 7.11962210492807880353203808353, 8.463143467690329197242557835324, 9.191379967491880023733018120069, 9.987475129535985256579168342141, 10.52809821899708824951318914210

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