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Properties

Label 2-252-7.6-c8-0-23
Degree $2$
Conductor $252$
Sign $-0.840 - 0.542i$
Analytic cond. $102.659$
Root an. cond. $10.1320$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.01e3 − 1.30e3i)7-s − 4.45e4i·13-s − 3.31e4i·19-s + 3.90e5·25-s − 3.70e5i·31-s − 5.03e5·37-s − 3.49e6·43-s + (2.37e6 + 5.25e6i)49-s + 1.41e7i·61-s − 5.42e6·67-s − 5.44e7i·73-s − 1.88e7·79-s + (−5.79e7 + 8.97e7i)91-s + 7.29e6i·97-s + 2.20e8i·103-s + ⋯
L(s)  = 1  + (−0.840 − 0.542i)7-s − 1.55i·13-s − 0.254i·19-s + 25-s − 0.401i·31-s − 0.268·37-s − 1.02·43-s + (0.411 + 0.911i)49-s + 1.01i·61-s − 0.269·67-s − 1.91i·73-s − 0.484·79-s + (−0.845 + 1.30i)91-s + 0.0824i·97-s + 1.96i·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 - 0.542i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.840 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.840 - 0.542i$
Analytic conductor: \(102.659\)
Root analytic conductor: \(10.1320\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :4),\ -0.840 - 0.542i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.1640619653\)
\(L(\frac12)\) \(\approx\) \(0.1640619653\)
\(L(5)\) 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 \( 1 \)
7 \( 1 + (2.01e3 + 1.30e3i)T \)
good5 \( 1 - 3.90e5T^{2} \)
11 \( 1 + 2.14e8T^{2} \)
13 \( 1 + 4.45e4iT - 8.15e8T^{2} \)
17 \( 1 - 6.97e9T^{2} \)
19 \( 1 + 3.31e4iT - 1.69e10T^{2} \)
23 \( 1 + 7.83e10T^{2} \)
29 \( 1 + 5.00e11T^{2} \)
31 \( 1 + 3.70e5iT - 8.52e11T^{2} \)
37 \( 1 + 5.03e5T + 3.51e12T^{2} \)
41 \( 1 - 7.98e12T^{2} \)
43 \( 1 + 3.49e6T + 1.16e13T^{2} \)
47 \( 1 - 2.38e13T^{2} \)
53 \( 1 + 6.22e13T^{2} \)
59 \( 1 - 1.46e14T^{2} \)
61 \( 1 - 1.41e7iT - 1.91e14T^{2} \)
67 \( 1 + 5.42e6T + 4.06e14T^{2} \)
71 \( 1 + 6.45e14T^{2} \)
73 \( 1 + 5.44e7iT - 8.06e14T^{2} \)
79 \( 1 + 1.88e7T + 1.51e15T^{2} \)
83 \( 1 - 2.25e15T^{2} \)
89 \( 1 - 3.93e15T^{2} \)
97 \( 1 - 7.29e6iT - 7.83e15T^{2} \)
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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

−10.19448133720106974784393131839, −9.145283720939465720375843709531, −8.036522164524983332997297716488, −7.07587436730270915155114787423, −6.06219614013097729955014785811, −4.94905359699931880326993744988, −3.60246206927458733317165858414, −2.72805724503235457614272130796, −1.02567421164257783272486034834, −0.03922643094772599394679203407, 1.54457392441546694783877535133, 2.75415922626583855686688768232, 3.92252966663567148008751348831, 5.13550455380447883697165988750, 6.36691980411738328873586772023, 7.02824461207092527601934001246, 8.484579868918358998129169867686, 9.244804172191471669208521935237, 10.09522703947783508740814517952, 11.26638787192662904128872324856

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