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Properties

Label 2-252-21.5-c7-0-13
Degree $2$
Conductor $252$
Sign $0.341 + 0.939i$
Analytic cond. $78.7210$
Root an. cond. $8.87248$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (21.6 − 37.4i)5-s + (487. + 765. i)7-s + (2.46e3 − 1.42e3i)11-s − 1.90e3i·13-s + (−1.71e4 − 2.97e4i)17-s + (−2.75e4 − 1.59e4i)19-s + (3.58e4 + 2.06e4i)23-s + (3.81e4 + 6.60e4i)25-s − 3.99e3i·29-s + (−1.10e5 + 6.39e4i)31-s + (3.92e4 − 1.70e3i)35-s + (2.43e5 − 4.21e5i)37-s − 1.19e4·41-s + 3.23e5·43-s + (−4.71e5 + 8.17e5i)47-s + ⋯
L(s)  = 1  + (0.0774 − 0.134i)5-s + (0.537 + 0.843i)7-s + (0.558 − 0.322i)11-s − 0.240i·13-s + (−0.848 − 1.46i)17-s + (−0.923 − 0.532i)19-s + (0.613 + 0.354i)23-s + (0.488 + 0.845i)25-s − 0.0304i·29-s + (−0.668 + 0.385i)31-s + (0.154 − 0.00673i)35-s + (0.790 − 1.36i)37-s − 0.0271·41-s + 0.619·43-s + (−0.662 + 1.14i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.341 + 0.939i$
Analytic conductor: \(78.7210\)
Root analytic conductor: \(8.87248\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :7/2),\ 0.341 + 0.939i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.879298642\)
\(L(\frac12)\) \(\approx\) \(1.879298642\)
\(L(\frac{9}{2})\) 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 \( 1 \)
7 \( 1 + (-487. - 765. i)T \)
good5 \( 1 + (-21.6 + 37.4i)T + (-3.90e4 - 6.76e4i)T^{2} \)
11 \( 1 + (-2.46e3 + 1.42e3i)T + (9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + 1.90e3iT - 6.27e7T^{2} \)
17 \( 1 + (1.71e4 + 2.97e4i)T + (-2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (2.75e4 + 1.59e4i)T + (4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (-3.58e4 - 2.06e4i)T + (1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + 3.99e3iT - 1.72e10T^{2} \)
31 \( 1 + (1.10e5 - 6.39e4i)T + (1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + (-2.43e5 + 4.21e5i)T + (-4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 + 1.19e4T + 1.94e11T^{2} \)
43 \( 1 - 3.23e5T + 2.71e11T^{2} \)
47 \( 1 + (4.71e5 - 8.17e5i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + (-1.78e6 + 1.02e6i)T + (5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (5.58e5 + 9.66e5i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (2.64e6 + 1.52e6i)T + (1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (5.02e5 + 8.70e5i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + 2.34e6iT - 9.09e12T^{2} \)
73 \( 1 + (-1.94e6 + 1.12e6i)T + (5.52e12 - 9.56e12i)T^{2} \)
79 \( 1 + (4.54e5 - 7.87e5i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 - 4.64e6T + 2.71e13T^{2} \)
89 \( 1 + (-4.97e6 + 8.61e6i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + 7.88e6iT - 8.07e13T^{2} \)
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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

−10.92415942631459992347089365712, −9.201985165235257596839476917142, −9.021284030162887655550099469961, −7.67584996562799836825615509789, −6.60300099640709277067383961371, −5.43873969106213135240888124938, −4.54993009567162287901109573810, −3.02697648887917276743922927789, −1.89801265151282306826895420482, −0.47064611695530770147848396880, 1.10166634157769611502831489800, 2.20446374423611244355238936414, 3.89924038352375647600232836076, 4.57084974631058641789185397313, 6.14955259945402330685263142052, 6.94517567960920581144319230097, 8.118176022043949140745005204405, 8.944307107173077261651582751497, 10.29650460531866653824023902663, 10.78810600702103762893257292968

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