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Properties

Label 2-252-28.19-c1-0-8
Degree $2$
Conductor $252$
Sign $-0.0206 - 0.999i$
Analytic cond. $2.01223$
Root an. cond. $1.41853$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.546 + 1.30i)2-s + (−1.40 + 1.42i)4-s + (3.33 + 1.92i)5-s + (1.59 − 2.11i)7-s + (−2.62 − 1.05i)8-s + (−0.690 + 5.40i)10-s + (−1.17 + 0.681i)11-s − 0.369i·13-s + (3.62 + 0.923i)14-s + (−0.0640 − 3.99i)16-s + (−3.89 + 2.25i)17-s + (0.0330 − 0.0573i)19-s + (−7.43 + 2.05i)20-s + (−1.53 − 1.16i)22-s + (−2.77 − 1.60i)23-s + ⋯
L(s)  = 1  + (0.386 + 0.922i)2-s + (−0.701 + 0.712i)4-s + (1.49 + 0.862i)5-s + (0.602 − 0.798i)7-s + (−0.928 − 0.371i)8-s + (−0.218 + 1.71i)10-s + (−0.355 + 0.205i)11-s − 0.102i·13-s + (0.969 + 0.246i)14-s + (−0.0160 − 0.999i)16-s + (−0.945 + 0.545i)17-s + (0.00759 − 0.0131i)19-s + (−1.66 + 0.459i)20-s + (−0.326 − 0.248i)22-s + (−0.579 − 0.334i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0206 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0206 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.0206 - 0.999i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.0206 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20713 + 1.23226i\)
\(L(\frac12)\) \(\approx\) \(1.20713 + 1.23226i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.546 - 1.30i)T \)
3 \( 1 \)
7 \( 1 + (-1.59 + 2.11i)T \)
good5 \( 1 + (-3.33 - 1.92i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.17 - 0.681i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.369iT - 13T^{2} \)
17 \( 1 + (3.89 - 2.25i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0330 + 0.0573i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.77 + 1.60i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.11T + 29T^{2} \)
31 \( 1 + (3.01 + 5.22i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.74 + 4.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.45iT - 41T^{2} \)
43 \( 1 - 6.30iT - 43T^{2} \)
47 \( 1 + (-0.712 + 1.23i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.27 + 2.20i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.71 + 2.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.23 - 0.715i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.45 + 4.88i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 + (1.56 - 0.900i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-10.8 - 6.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 + (1.11 + 0.646i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.88iT - 97T^{2} \)
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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

−12.74458912635176172508687065677, −11.16646835554708040244103987994, −10.30194260995437805801122580071, −9.400296862882507066037515033964, −8.166542560953255466980238549432, −7.09439272930395383635835611720, −6.29582225986796021712796578856, −5.32286498477727899526715202230, −4.03846348595522515592845765906, −2.33264728578967249796656254934, 1.59156030031612830016084976914, 2.64606448723190940849797825502, 4.66021474433818028940933093985, 5.36372294756156158350282004443, 6.28026468404229220751376426656, 8.399884273484248906639699812810, 9.125360955380085250363321273301, 9.874625421860904845123778872811, 10.91030760638101758839928579398, 11.91848709016409277883152003103

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