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Properties

Label 2-252-7.2-c3-0-3
Degree $2$
Conductor $252$
Sign $0.723 - 0.690i$
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  + (−6.41 + 11.1i)5-s + (6.32 − 17.4i)7-s + (−18.4 − 31.8i)11-s + 87.1·13-s + (51.2 + 88.8i)17-s + (−47.9 + 82.9i)19-s + (−48 + 83.1i)23-s + (−19.7 − 34.1i)25-s + 212.·29-s + (79.6 + 137. i)31-s + (152. + 181. i)35-s + (−64.3 + 111. i)37-s + 298.·41-s − 33.3·43-s + (135. − 234. i)47-s + ⋯
L(s)  = 1  + (−0.573 + 0.993i)5-s + (0.341 − 0.939i)7-s + (−0.504 − 0.874i)11-s + 1.85·13-s + (0.731 + 1.26i)17-s + (−0.578 + 1.00i)19-s + (−0.435 + 0.753i)23-s + (−0.157 − 0.273i)25-s + 1.35·29-s + (0.461 + 0.799i)31-s + (0.737 + 0.878i)35-s + (−0.285 + 0.495i)37-s + 1.13·41-s − 0.118·43-s + (0.420 − 0.728i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.723 - 0.690i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.723 - 0.690i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.684658976\)
\(L(\frac12)\) \(\approx\) \(1.684658976\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 + (-6.32 + 17.4i)T \)
good5 \( 1 + (6.41 - 11.1i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (18.4 + 31.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 87.1T + 2.19e3T^{2} \)
17 \( 1 + (-51.2 - 88.8i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (47.9 - 82.9i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (48 - 83.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 212.T + 2.43e4T^{2} \)
31 \( 1 + (-79.6 - 137. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (64.3 - 111. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 298.T + 6.89e4T^{2} \)
43 \( 1 + 33.3T + 7.95e4T^{2} \)
47 \( 1 + (-135. + 234. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-224. - 388. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (334. + 578. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-121. + 211. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-167. - 290. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 339.T + 3.57e5T^{2} \)
73 \( 1 + (-459. - 795. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-68.1 + 118. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 287.T + 5.71e5T^{2} \)
89 \( 1 + (-80.9 + 140. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 182.T + 9.12e5T^{2} \)
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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

−11.42087154898875919365523938707, −10.71193782678460116594906511881, −10.27739738029949189745095937104, −8.389670027136311284370071343911, −7.984195033317557187493964641075, −6.67541479118251028953386166663, −5.80870439502315264124942488377, −3.96416539822846191491645940977, −3.33795141277712125428586543473, −1.23197585453533349326073923669, 0.823594089859419767699424459504, 2.55433378406029226241405310717, 4.28055608045716457502284919448, 5.13693794376654035208045225140, 6.34372779517028820917989802959, 7.80675738459040371438924046372, 8.576915495538355368034919589583, 9.297742384700761443751485349507, 10.64356955305887291278174028973, 11.67681807776979663869300378809

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