Properties

Label 2-252-7.2-c3-0-2
Degree $2$
Conductor $252$
Sign $-0.0165 - 0.999i$
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  + (8.5 + 16.4i)7-s − 19·13-s + (−53.5 + 92.6i)19-s + (62.5 + 108. i)25-s + (144.5 + 250. i)31-s + (−161.5 + 279. i)37-s + 71·43-s + (−198.5 + 279. i)49-s + (−91 + 157. i)61-s + (63.5 + 109. i)67-s + (135.5 + 234. i)73-s + (693.5 − 1.20e3i)79-s + (−161.5 − 312. i)91-s − 1.33e3·97-s + (900.5 − 1.55e3i)103-s + ⋯
L(s)  = 1  + (0.458 + 0.888i)7-s − 0.405·13-s + (−0.645 + 1.11i)19-s + (0.5 + 0.866i)25-s + (0.837 + 1.45i)31-s + (−0.717 + 1.24i)37-s + 0.251·43-s + (−0.578 + 0.815i)49-s + (−0.191 + 0.330i)61-s + (0.115 + 0.200i)67-s + (0.217 + 0.376i)73-s + (0.987 − 1.71i)79-s + (−0.186 − 0.360i)91-s − 1.39·97-s + (0.861 − 1.49i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0165 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0165 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.0165 - 0.999i$
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: \((2,\ 252,\ (\ :3/2),\ -0.0165 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.483523752\)
\(L(\frac12)\) \(\approx\) \(1.483523752\)
\(L(\frac{5}{2})\) 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 \( 1 + (-8.5 - 16.4i)T \)
good5 \( 1 + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 19T + 2.19e3T^{2} \)
17 \( 1 + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (53.5 - 92.6i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 2.43e4T^{2} \)
31 \( 1 + (-144.5 - 250. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (161.5 - 279. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 - 71T + 7.95e4T^{2} \)
47 \( 1 + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (91 - 157. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-63.5 - 109. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 + (-135.5 - 234. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-693.5 + 1.20e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.33e3T + 9.12e5T^{2} \)
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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

−11.99142862389887593208134056956, −10.90418008308671326571106410172, −9.949373293350446087258603932355, −8.827910867186952998273668740303, −8.089563139334207279806188194282, −6.82138810938263262783923667325, −5.66614437925952831607898670657, −4.66329370800323904244372253302, −3.09199133555047477090152710617, −1.65308511388879026424102448556, 0.59445987539238523480168153030, 2.37165980107151361586766339429, 4.02240455283875516625804132796, 4.96848099143220736352785486481, 6.43273735845060132822148579398, 7.39244879268777096048117576275, 8.358715037796500356309811049923, 9.487188821549907607488293309340, 10.52499942711777554767940032264, 11.22188209352479935401790272884

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