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Properties

Label 2-252-63.59-c3-0-4
Degree $2$
Conductor $252$
Sign $0.109 - 0.993i$
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  + (−3.15 − 4.12i)3-s + 18.3·5-s + (−11.4 + 14.5i)7-s + (−7.09 + 26.0i)9-s + 55.3i·11-s + (−68.0 − 39.3i)13-s + (−57.9 − 75.8i)15-s + (−9.99 + 17.3i)17-s + (−16.7 + 9.67i)19-s + (96.2 + 1.61i)21-s + 124. i·23-s + 212.·25-s + (129. − 52.9i)27-s + (−48.2 + 27.8i)29-s + (−275. + 158. i)31-s + ⋯
L(s)  = 1  + (−0.607 − 0.794i)3-s + 1.64·5-s + (−0.620 + 0.784i)7-s + (−0.262 + 0.964i)9-s + 1.51i·11-s + (−1.45 − 0.838i)13-s + (−0.998 − 1.30i)15-s + (−0.142 + 0.246i)17-s + (−0.202 + 0.116i)19-s + (0.999 + 0.0167i)21-s + 1.12i·23-s + 1.70·25-s + (0.926 − 0.377i)27-s + (−0.309 + 0.178i)29-s + (−1.59 + 0.921i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.109 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.109 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.109 - 0.993i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.109 - 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.139262210\)
\(L(\frac12)\) \(\approx\) \(1.139262210\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.15 + 4.12i)T \)
7 \( 1 + (11.4 - 14.5i)T \)
good5 \( 1 - 18.3T + 125T^{2} \)
11 \( 1 - 55.3iT - 1.33e3T^{2} \)
13 \( 1 + (68.0 + 39.3i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (9.99 - 17.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (16.7 - 9.67i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 - 124. iT - 1.21e4T^{2} \)
29 \( 1 + (48.2 - 27.8i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (275. - 158. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-20.5 - 35.5i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-53.1 + 92.0i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-279. - 483. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-73.9 + 128. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-570. - 329. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (284. + 492. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (122. + 70.4i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-147. - 254. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 602. iT - 3.57e5T^{2} \)
73 \( 1 + (-541. - 312. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-267. + 462. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-352. - 610. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-38.6 - 66.9i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (382. - 220. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.29905482669282048331624839420, −10.74282285939044689762513398855, −9.824484087720722250967748454509, −9.260805488296419255317904751878, −7.59329034159678733595558970183, −6.73274848426855994593449817614, −5.66828033215835157236446055826, −5.10515159692673733917424385175, −2.56421443632642394086407683013, −1.75971006636211918090431074525, 0.45361476305796952552300525338, 2.50023429148973210450201532814, 4.03112450279819385919866213023, 5.33904385554426677389042120531, 6.12600039291662972263140651807, 7.03807303257599867051665491133, 8.992590150005569636854168954139, 9.533972747192711562481223602810, 10.40083325275088784132996352128, 11.04757525477291110830571139991

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