# Properties

 Label 2-252-63.20-c3-0-13 Degree $2$ Conductor $252$ Sign $0.900 + 0.435i$ Analytic cond. $14.8684$ Root an. cond. $3.85596$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (4.07 − 3.21i)3-s + (−0.330 − 0.571i)5-s + (−0.762 + 18.5i)7-s + (6.26 − 26.2i)9-s + (21.4 + 12.3i)11-s + (43.5 − 25.1i)13-s + (−3.18 − 1.26i)15-s + 67.5·17-s + 62.9i·19-s + (56.4 + 77.9i)21-s + (135. − 78.4i)23-s + (62.2 − 107. i)25-s + (−58.9 − 127. i)27-s + (−129. − 74.9i)29-s + (−139. + 80.5i)31-s + ⋯
 L(s)  = 1 + (0.784 − 0.619i)3-s + (−0.0295 − 0.0511i)5-s + (−0.0411 + 0.999i)7-s + (0.232 − 0.972i)9-s + (0.586 + 0.338i)11-s + (0.928 − 0.536i)13-s + (−0.0548 − 0.0218i)15-s + 0.963·17-s + 0.760i·19-s + (0.586 + 0.809i)21-s + (1.23 − 0.710i)23-s + (0.498 − 0.863i)25-s + (−0.420 − 0.907i)27-s + (−0.831 − 0.480i)29-s + (−0.808 + 0.466i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$252$$    =    $$2^{2} \cdot 3^{2} \cdot 7$$ Sign: $0.900 + 0.435i$ Analytic conductor: $$14.8684$$ Root analytic conductor: $$3.85596$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{252} (209, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 252,\ (\ :3/2),\ 0.900 + 0.435i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.558098079$$ $$L(\frac12)$$ $$\approx$$ $$2.558098079$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (-4.07 + 3.21i)T$$
7 $$1 + (0.762 - 18.5i)T$$
good5 $$1 + (0.330 + 0.571i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (-21.4 - 12.3i)T + (665.5 + 1.15e3i)T^{2}$$
13 $$1 + (-43.5 + 25.1i)T + (1.09e3 - 1.90e3i)T^{2}$$
17 $$1 - 67.5T + 4.91e3T^{2}$$
19 $$1 - 62.9iT - 6.85e3T^{2}$$
23 $$1 + (-135. + 78.4i)T + (6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + (129. + 74.9i)T + (1.21e4 + 2.11e4i)T^{2}$$
31 $$1 + (139. - 80.5i)T + (1.48e4 - 2.57e4i)T^{2}$$
37 $$1 + 16.2T + 5.06e4T^{2}$$
41 $$1 + (-134. - 233. i)T + (-3.44e4 + 5.96e4i)T^{2}$$
43 $$1 + (-188. + 325. i)T + (-3.97e4 - 6.88e4i)T^{2}$$
47 $$1 + (-31.3 + 54.3i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 - 136. iT - 1.48e5T^{2}$$
59 $$1 + (-358. - 621. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (23.9 + 13.8i)T + (1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (163. + 283. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 - 246. iT - 3.57e5T^{2}$$
73 $$1 - 261. iT - 3.89e5T^{2}$$
79 $$1 + (391. - 678. i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + (-599. + 1.03e3i)T + (-2.85e5 - 4.95e5i)T^{2}$$
89 $$1 - 968.T + 7.04e5T^{2}$$
97 $$1 + (1.10e3 + 639. i)T + (4.56e5 + 7.90e5i)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

−11.84593477063427259498709156912, −10.51905403154352172847156014257, −9.297287110196812387728446201693, −8.646543255192291057828481306638, −7.72826669138941590439277158336, −6.54700742324600345922069931517, −5.55564511541789265853748520242, −3.81952740091640502227362656767, −2.65218344568240555649685640401, −1.22509484052628473307464571763, 1.32338048968007571823023898543, 3.26045035180330717764085895725, 4.00442961400311087138628705499, 5.32736012584330190617181088177, 6.89690292311828593987841490536, 7.74817059998689655209846965593, 9.001518730588086927246728439280, 9.524448230059395774029657507331, 10.87382592600126278694840589383, 11.21256146401602197657921980693