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Properties

Label 2-252-63.4-c1-0-4
Degree $2$
Conductor $252$
Sign $0.591 + 0.806i$
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  + (−1.73 + 0.0755i)3-s + 0.967·5-s + (−1.11 − 2.39i)7-s + (2.98 − 0.261i)9-s + 0.728·11-s + (1.81 − 3.13i)13-s + (−1.67 + 0.0731i)15-s + (3.49 − 6.06i)17-s + (−0.348 − 0.602i)19-s + (2.11 + 4.06i)21-s + 6.43·23-s − 4.06·25-s + (−5.15 + 0.678i)27-s + (3.34 + 5.79i)29-s + (−4.58 − 7.93i)31-s + ⋯
L(s)  = 1  + (−0.999 + 0.0436i)3-s + 0.432·5-s + (−0.421 − 0.906i)7-s + (0.996 − 0.0871i)9-s + 0.219·11-s + (0.502 − 0.869i)13-s + (−0.432 + 0.0188i)15-s + (0.848 − 1.47i)17-s + (−0.0798 − 0.138i)19-s + (0.460 + 0.887i)21-s + 1.34·23-s − 0.812·25-s + (−0.991 + 0.130i)27-s + (0.621 + 1.07i)29-s + (−0.823 − 1.42i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.591 + 0.806i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.591 + 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.849046 - 0.429899i\)
\(L(\frac12)\) \(\approx\) \(0.849046 - 0.429899i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.73 - 0.0755i)T \)
7 \( 1 + (1.11 + 2.39i)T \)
good5 \( 1 - 0.967T + 5T^{2} \)
11 \( 1 - 0.728T + 11T^{2} \)
13 \( 1 + (-1.81 + 3.13i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.49 + 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.348 + 0.602i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.43T + 23T^{2} \)
29 \( 1 + (-3.34 - 5.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.58 + 7.93i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.854 - 1.48i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.62 - 6.27i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.348 + 0.602i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.83 - 6.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.05 + 3.56i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.38 - 4.13i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.46 - 4.26i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.91 - 5.04i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.304T + 71T^{2} \)
73 \( 1 + (-5.33 + 9.24i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.61 + 2.80i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.618 - 1.07i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.78 + 10.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.32 - 2.30i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.76263744332934466991528230743, −10.94969933936571337010668397335, −10.08756752406077443544936129710, −9.355796243924277447481477088888, −7.68090796045734067604852681986, −6.82890688452275551310932509258, −5.77520833437970793139994376194, −4.77255696217525335687639721039, −3.31132625728542440156023607267, −0.947372097348023897407639785286, 1.74720809653009538082668078772, 3.74077476252067360009539968370, 5.24654755940994269454245048943, 6.07850862716383830425586936980, 6.86196979583613177258245789628, 8.402146173457391672636717695926, 9.441856767557045195707026080370, 10.33868483159964528251304927286, 11.30650393960380466688253081234, 12.21254050102996806606519531564

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