Properties

Label 2-252-7.4-c7-0-22
Degree $2$
Conductor $252$
Sign $-0.677 - 0.735i$
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  + (−80.0 − 138. i)5-s + (−254. − 871. i)7-s + (1.58e3 − 2.73e3i)11-s − 4.77e3·13-s + (−6.50e3 + 1.12e4i)17-s + (−2.20e4 − 3.82e4i)19-s + (−3.25e4 − 5.63e4i)23-s + (2.62e4 − 4.54e4i)25-s + 2.46e5·29-s + (−1.50e5 + 2.60e5i)31-s + (−1.00e5 + 1.05e5i)35-s + (−2.58e5 − 4.47e5i)37-s + 3.77e5·41-s − 7.14e4·43-s + (5.58e5 + 9.67e5i)47-s + ⋯
L(s)  = 1  + (−0.286 − 0.496i)5-s + (−0.280 − 0.959i)7-s + (0.358 − 0.620i)11-s − 0.602·13-s + (−0.321 + 0.556i)17-s + (−0.737 − 1.27i)19-s + (−0.557 − 0.965i)23-s + (0.335 − 0.581i)25-s + 1.87·29-s + (−0.906 + 1.56i)31-s + (−0.395 + 0.414i)35-s + (−0.838 − 1.45i)37-s + 0.856·41-s − 0.136·43-s + (0.784 + 1.35i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.677 - 0.735i$
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: \((2,\ 252,\ (\ :7/2),\ -0.677 - 0.735i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.3176540593\)
\(L(\frac12)\) \(\approx\) \(0.3176540593\)
\(L(\frac{9}{2})\) 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 \( 1 + (254. + 871. i)T \)
good5 \( 1 + (80.0 + 138. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
11 \( 1 + (-1.58e3 + 2.73e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + 4.77e3T + 6.27e7T^{2} \)
17 \( 1 + (6.50e3 - 1.12e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (2.20e4 + 3.82e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (3.25e4 + 5.63e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 - 2.46e5T + 1.72e10T^{2} \)
31 \( 1 + (1.50e5 - 2.60e5i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + (2.58e5 + 4.47e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 - 3.77e5T + 1.94e11T^{2} \)
43 \( 1 + 7.14e4T + 2.71e11T^{2} \)
47 \( 1 + (-5.58e5 - 9.67e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (1.84e5 - 3.19e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-5.15e5 + 8.93e5i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-1.61e5 - 2.79e5i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-1.17e4 + 2.03e4i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + 2.84e6T + 9.09e12T^{2} \)
73 \( 1 + (3.36e5 - 5.82e5i)T + (-5.52e12 - 9.56e12i)T^{2} \)
79 \( 1 + (-4.29e5 - 7.43e5i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + 7.32e6T + 2.71e13T^{2} \)
89 \( 1 + (-2.31e6 - 4.00e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 - 9.51e5T + 8.07e13T^{2} \)
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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

−10.41517243770313656111787369057, −9.062174553694641379195996204280, −8.371224988451845938915271662304, −7.12501081837938691679385342931, −6.32500758161122502486971420578, −4.79065791493555708095756699318, −4.02941160619509091429328695712, −2.64415035682650806199469710945, −1.00327583645059533311531871694, −0.082667881330453216909519311416, 1.79181477035950271719873405015, 2.86970790358564565307432731827, 4.11640199178377804158937536517, 5.40277357020348659427779617336, 6.46507389596911090519745351072, 7.43316624707313081510932717256, 8.518225566511710669785563627972, 9.563627339598153489686795659364, 10.31716985145336871664149009997, 11.64585394054968346294205795704

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