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Properties

Label 2-252-252.103-c1-0-30
Degree $2$
Conductor $252$
Sign $0.871 - 0.490i$
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  + (1.27 + 0.603i)2-s + (1.58 − 0.689i)3-s + (1.27 + 1.54i)4-s + (−0.0437 + 0.0252i)5-s + (2.44 + 0.0774i)6-s + (−1.31 + 2.29i)7-s + (0.693 + 2.74i)8-s + (2.04 − 2.19i)9-s + (−0.0712 + 0.00589i)10-s + (−4.52 − 2.61i)11-s + (3.08 + 1.57i)12-s + (−3.10 − 1.79i)13-s + (−3.06 + 2.14i)14-s + (−0.0521 + 0.0703i)15-s + (−0.768 + 3.92i)16-s + (3.76 − 2.17i)17-s + ⋯
L(s)  = 1  + (0.904 + 0.426i)2-s + (0.917 − 0.398i)3-s + (0.635 + 0.772i)4-s + (−0.0195 + 0.0113i)5-s + (0.999 + 0.0316i)6-s + (−0.496 + 0.868i)7-s + (0.245 + 0.969i)8-s + (0.683 − 0.730i)9-s + (−0.0225 + 0.00186i)10-s + (−1.36 − 0.788i)11-s + (0.890 + 0.455i)12-s + (−0.860 − 0.497i)13-s + (−0.819 + 0.573i)14-s + (−0.0134 + 0.0181i)15-s + (−0.192 + 0.981i)16-s + (0.912 − 0.526i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.871 - 0.490i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.871 - 0.490i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.36805 + 0.620335i\)
\(L(\frac12)\) \(\approx\) \(2.36805 + 0.620335i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.27 - 0.603i)T \)
3 \( 1 + (-1.58 + 0.689i)T \)
7 \( 1 + (1.31 - 2.29i)T \)
good5 \( 1 + (0.0437 - 0.0252i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.52 + 2.61i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.10 + 1.79i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.76 + 2.17i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.88 + 3.27i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.66 + 0.958i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.87 + 3.24i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.776T + 31T^{2} \)
37 \( 1 + (5.64 - 9.77i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.98 - 3.45i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.0488 - 0.0282i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.34T + 47T^{2} \)
53 \( 1 + (-0.804 - 1.39i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 7.66T + 59T^{2} \)
61 \( 1 - 9.67iT - 61T^{2} \)
67 \( 1 - 9.49iT - 67T^{2} \)
71 \( 1 + 10.8iT - 71T^{2} \)
73 \( 1 + (-3.81 + 2.20i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 3.45iT - 79T^{2} \)
83 \( 1 + (5.85 + 10.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.18 + 2.41i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.35 + 0.781i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.42901151857673716482471848108, −11.58441247291003395943542110360, −10.13512148573062451317494420998, −8.985291897558681649018996839397, −7.932196255089316235497848818674, −7.29010794011950731777237300950, −5.93020229173390119301602269410, −5.01434064666061615102109592949, −3.18020415473313274536615977693, −2.64812364495413517643320727298, 2.12476364124338225651148802242, 3.39385103167558063565751273380, 4.38324259238730976042591861623, 5.48130613997289740606409958208, 7.16055404231161125177043526116, 7.75526859929297474974936988319, 9.511499010629105833843553657009, 10.16045632368141830897924957939, 10.77549688673070319815100639566, 12.43735711318086628969565402533

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