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Properties

Label 2-252-63.41-c3-0-14
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 $0$

Origins

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Normalization:  

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: $\chi_{252} (41, \cdot )$
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

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

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