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Properties

Label 2-252-36.23-c1-0-27
Degree $2$
Conductor $252$
Sign $-0.0380 + 0.999i$
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  + (−1.16 + 0.801i)2-s + (1.63 − 0.574i)3-s + (0.713 − 1.86i)4-s + (−3.32 − 1.91i)5-s + (−1.44 + 1.97i)6-s + (−0.866 + 0.5i)7-s + (0.667 + 2.74i)8-s + (2.34 − 1.87i)9-s + (5.41 − 0.430i)10-s + (−2.92 − 5.06i)11-s + (0.0932 − 3.46i)12-s + (−0.515 + 0.892i)13-s + (0.607 − 1.27i)14-s + (−6.53 − 1.22i)15-s + (−2.98 − 2.66i)16-s − 0.189i·17-s + ⋯
L(s)  = 1  + (−0.823 + 0.567i)2-s + (0.943 − 0.331i)3-s + (0.356 − 0.934i)4-s + (−1.48 − 0.858i)5-s + (−0.589 + 0.808i)6-s + (−0.327 + 0.188i)7-s + (0.235 + 0.971i)8-s + (0.780 − 0.625i)9-s + (1.71 − 0.136i)10-s + (−0.881 − 1.52i)11-s + (0.0269 − 0.999i)12-s + (−0.142 + 0.247i)13-s + (0.162 − 0.341i)14-s + (−1.68 − 0.316i)15-s + (−0.745 − 0.666i)16-s − 0.0460i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0380 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0380 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.0380 + 0.999i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.0380 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.478409 - 0.496957i\)
\(L(\frac12)\) \(\approx\) \(0.478409 - 0.496957i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 - 0.801i)T \)
3 \( 1 + (-1.63 + 0.574i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (3.32 + 1.91i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.92 + 5.06i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.515 - 0.892i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.189iT - 17T^{2} \)
19 \( 1 + 4.14iT - 19T^{2} \)
23 \( 1 + (0.717 - 1.24i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.21 + 1.85i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.05 + 3.49i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.78T + 37T^{2} \)
41 \( 1 + (-8.15 - 4.70i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.25 + 3.03i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.78 - 3.08i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.68iT - 53T^{2} \)
59 \( 1 + (4.24 - 7.34i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.477 + 0.826i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.704 + 0.406i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.09T + 71T^{2} \)
73 \( 1 + 8.12T + 73T^{2} \)
79 \( 1 + (-6.10 + 3.52i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.57 + 13.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.51iT - 89T^{2} \)
97 \( 1 + (-1.26 - 2.19i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.67661053918952962911524066195, −10.83684494774248526195690570358, −9.329546606716465655328932975117, −8.744009002592677541504857153879, −7.934711715628419268001305732359, −7.37260479759897367420664607798, −5.90166606889955797618952171967, −4.39759105774864215762886978415, −2.88093528307257297805640401542, −0.63258562135300450406200069313, 2.40836079104280930296253826361, 3.48591213699430166039477976723, 4.39191784102678837061553918107, 7.04960342583501276644624948728, 7.58882964280082187508795235827, 8.276146732635316725485151490635, 9.568915371035967685869819332629, 10.36985412098620092810891370572, 10.98031565244011487386374472338, 12.35899823887149578724714364698

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