Properties

Label 2-252-21.2-c8-0-18
Degree $2$
Conductor $252$
Sign $-0.251 + 0.967i$
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  + (674. + 389. i)5-s + (−1.63e3 + 1.75e3i)7-s + (−291. + 168. i)11-s + 46.7·13-s + (−9.81e4 + 5.66e4i)17-s + (4.11e4 − 7.13e4i)19-s + (−3.26e5 − 1.88e5i)23-s + (1.07e5 + 1.87e5i)25-s + 7.47e5i·29-s + (−7.74e5 − 1.34e6i)31-s + (−1.78e6 + 5.49e5i)35-s + (4.75e5 − 8.23e5i)37-s − 3.98e6i·41-s − 1.78e5·43-s + (4.60e6 + 2.66e6i)47-s + ⋯
L(s)  = 1  + (1.07 + 0.623i)5-s + (−0.680 + 0.732i)7-s + (−0.0199 + 0.0115i)11-s + 0.00163·13-s + (−1.17 + 0.678i)17-s + (0.316 − 0.547i)19-s + (−1.16 − 0.673i)23-s + (0.276 + 0.478i)25-s + 1.05i·29-s + (−0.838 − 1.45i)31-s + (−1.19 + 0.366i)35-s + (0.253 − 0.439i)37-s − 1.41i·41-s − 0.0523·43-s + (0.944 + 0.545i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.251 + 0.967i$
Analytic conductor: \(102.659\)
Root analytic conductor: \(10.1320\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :4),\ -0.251 + 0.967i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.6970671515\)
\(L(\frac12)\) \(\approx\) \(0.6970671515\)
\(L(5)\) 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 \( 1 + (1.63e3 - 1.75e3i)T \)
good5 \( 1 + (-674. - 389. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (291. - 168. i)T + (1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 - 46.7T + 8.15e8T^{2} \)
17 \( 1 + (9.81e4 - 5.66e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (-4.11e4 + 7.13e4i)T + (-8.49e9 - 1.47e10i)T^{2} \)
23 \( 1 + (3.26e5 + 1.88e5i)T + (3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 - 7.47e5iT - 5.00e11T^{2} \)
31 \( 1 + (7.74e5 + 1.34e6i)T + (-4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 + (-4.75e5 + 8.23e5i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 + 3.98e6iT - 7.98e12T^{2} \)
43 \( 1 + 1.78e5T + 1.16e13T^{2} \)
47 \( 1 + (-4.60e6 - 2.66e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (-7.08e5 + 4.09e5i)T + (3.11e13 - 5.39e13i)T^{2} \)
59 \( 1 + (-8.19e6 + 4.72e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (1.08e7 - 1.87e7i)T + (-9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (-7.78e6 - 1.34e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 + 7.95e6iT - 6.45e14T^{2} \)
73 \( 1 + (-1.59e7 - 2.75e7i)T + (-4.03e14 + 6.98e14i)T^{2} \)
79 \( 1 + (-2.44e7 + 4.23e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + 9.26e6iT - 2.25e15T^{2} \)
89 \( 1 + (3.56e7 + 2.05e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 + 6.33e7T + 7.83e15T^{2} \)
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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

−10.28800222860595830771575998967, −9.404120144340305452352712435558, −8.649600485975053337018297867221, −7.14649757329820727311837133183, −6.22101629995065747692960031900, −5.58750359086527514149624225475, −4.02703776719189731176078480288, −2.61927419156904250366870582281, −1.98201855502651119226802709561, −0.14431223891684396650387398873, 1.12612012441197187164270157833, 2.23676311054981734249092429424, 3.64793834528962798537735880179, 4.84987334049250444198169493257, 5.92441557207941275766861795844, 6.80607417792898111453863129285, 7.999376371265666205012686763370, 9.271644463122314746355598141144, 9.756508945444854082250389485108, 10.71351375209328319858660918213

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