# 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 $0$

# Related objects

## 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}$$
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$$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

−11.45344480480780197432930795485, −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