↑

Properties

Label 2-315-7.4-c3-0-16
Degree $2$
Conductor $315$
Sign $-0.266 - 0.963i$
Analytic cond. $18.5856$
Root an. cond. $4.31110$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 2.59i)2-s + (−0.5 + 0.866i)4-s + (2.5 + 4.33i)5-s + (−14 − 12.1i)7-s + 21·8-s + (−7.50 + 12.9i)10-s + (−22.5 + 38.9i)11-s + 59·13-s + (10.5 − 54.5i)14-s + (35.5 + 61.4i)16-s + (−27 + 46.7i)17-s + (60.5 + 104. i)19-s − 5.00·20-s − 135·22-s + (34.5 + 59.7i)23-s + ⋯
L(s)  = 1  + (0.530 + 0.918i)2-s + (−0.0625 + 0.108i)4-s + (0.223 + 0.387i)5-s + (−0.755 − 0.654i)7-s + 0.928·8-s + (−0.237 + 0.410i)10-s + (−0.616 + 1.06i)11-s + 1.25·13-s + (0.200 − 1.04i)14-s + (0.554 + 0.960i)16-s + (−0.385 + 0.667i)17-s + (0.730 + 1.26i)19-s − 0.0559·20-s − 1.30·22-s + (0.312 + 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.266 - 0.963i$
Analytic conductor: \(18.5856\)
Root analytic conductor: \(4.31110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :3/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.565231348\)
\(L(\frac12)\) \(\approx\) \(2.565231348\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.5 - 4.33i)T \)
7 \( 1 + (14 + 12.1i)T \)
good2 \( 1 + (-1.5 - 2.59i)T + (-4 + 6.92i)T^{2} \)
11 \( 1 + (22.5 - 38.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 59T + 2.19e3T^{2} \)
17 \( 1 + (27 - 46.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-60.5 - 104. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-34.5 - 59.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 162T + 2.43e4T^{2} \)
31 \( 1 + (-44 + 76.2i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-129.5 - 224. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 195T + 6.89e4T^{2} \)
43 \( 1 + 286T + 7.95e4T^{2} \)
47 \( 1 + (-22.5 - 38.9i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-298.5 + 517. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (180 - 311. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (196 + 339. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-140 + 242. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 48T + 3.57e5T^{2} \)
73 \( 1 + (334 - 578. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (391 + 677. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 768T + 5.71e5T^{2} \)
89 \( 1 + (597 + 1.03e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 902T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.45344480480780197432930795485, −10.24593016683928950709509079292, −10.01902761669125130777651336784, −8.326285987154752918201541311503, −7.37064360620068434728858112806, −6.54002760955191148778777047398, −5.81579668198679918505871562914, −4.55306617318921454797785601829, −3.41299952749232803619568782947, −1.54048225601424233423591348144, 0.847173441393604463168246249048, 2.59826281106166696326709631854, 3.30621967978097006843078618470, 4.72263679836319394521660015244, 5.78033631631063722413968915196, 6.93323746119566942113684055361, 8.382529561121609237670508893506, 9.090601050003599218191008792564, 10.30637304654184457781532351652, 11.15339304792633964524612685775

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