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Properties

Label 2-252-7.4-c7-0-1
Degree $2$
Conductor $252$
Sign $-0.980 + 0.196i$
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  + (194. + 336. i)5-s + (−876. − 235. i)7-s + (509. − 883. i)11-s + 2.94e3·13-s + (−3.44e3 + 5.96e3i)17-s + (1.22e4 + 2.12e4i)19-s + (−4.39e3 − 7.60e3i)23-s + (−3.64e4 + 6.30e4i)25-s + 7.96e3·29-s + (−1.60e5 + 2.77e5i)31-s + (−9.10e4 − 3.40e5i)35-s + (4.27e4 + 7.40e4i)37-s − 5.71e5·41-s + 1.43e5·43-s + (−3.38e5 − 5.86e5i)47-s + ⋯
L(s)  = 1  + (0.695 + 1.20i)5-s + (−0.965 − 0.259i)7-s + (0.115 − 0.200i)11-s + 0.371·13-s + (−0.169 + 0.294i)17-s + (0.411 + 0.712i)19-s + (−0.0752 − 0.130i)23-s + (−0.466 + 0.807i)25-s + 0.0606·29-s + (−0.965 + 1.67i)31-s + (−0.358 − 1.34i)35-s + (0.138 + 0.240i)37-s − 1.29·41-s + 0.275·43-s + (−0.475 − 0.823i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.196i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.980 + 0.196i$
Analytic conductor: \(78.7210\)
Root analytic conductor: \(8.87248\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :7/2),\ -0.980 + 0.196i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.5709780196\)
\(L(\frac12)\) \(\approx\) \(0.5709780196\)
\(L(\frac{9}{2})\) 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 \( 1 + (876. + 235. i)T \)
good5 \( 1 + (-194. - 336. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
11 \( 1 + (-509. + 883. i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 - 2.94e3T + 6.27e7T^{2} \)
17 \( 1 + (3.44e3 - 5.96e3i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (-1.22e4 - 2.12e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (4.39e3 + 7.60e3i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 - 7.96e3T + 1.72e10T^{2} \)
31 \( 1 + (1.60e5 - 2.77e5i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + (-4.27e4 - 7.40e4i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + 5.71e5T + 1.94e11T^{2} \)
43 \( 1 - 1.43e5T + 2.71e11T^{2} \)
47 \( 1 + (3.38e5 + 5.86e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (4.11e4 - 7.13e4i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-1.26e6 + 2.19e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (4.20e5 + 7.28e5i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-1.10e6 + 1.91e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + 1.63e6T + 9.09e12T^{2} \)
73 \( 1 + (2.09e6 - 3.63e6i)T + (-5.52e12 - 9.56e12i)T^{2} \)
79 \( 1 + (1.73e6 + 3.00e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + 8.82e6T + 2.71e13T^{2} \)
89 \( 1 + (-2.58e6 - 4.47e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + 1.34e7T + 8.07e13T^{2} \)
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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

−11.05140218146391569300691356774, −10.31236563584744120487924598900, −9.648926690473251621807451805621, −8.450394428539067383356496794782, −7.03292936132240950471953356752, −6.48325258943780753362999688984, −5.48519058773169409002567752017, −3.70801476207716751754005918110, −2.94353645330222181909802782761, −1.59360927568734609810565129808, 0.12818363801896014559352687830, 1.32269035525265063854496139975, 2.62819995060171954256372537433, 4.06651693859808295164186031989, 5.27537126071092081942945372422, 6.07654696043166602461056356307, 7.25895026720436666877129552458, 8.645017336601761792717726204071, 9.321810947848119891438418487259, 9.973620061306633457874190633963

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