Properties

Label 2-252-12.11-c7-0-23
Degree $2$
Conductor $252$
Sign $0.914 + 0.405i$
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  + (−1.11 − 11.2i)2-s + (−125. + 25.1i)4-s + 183. i·5-s − 343i·7-s + (423. + 1.38e3i)8-s + (2.06e3 − 205. i)10-s − 5.11e3·11-s − 1.12e4·13-s + (−3.86e3 + 383. i)14-s + (1.51e4 − 6.32e3i)16-s − 5.65e3i·17-s − 5.02e4i·19-s + (−4.62e3 − 2.30e4i)20-s + (5.71e3 + 5.75e4i)22-s − 5.34e4·23-s + ⋯
L(s)  = 1  + (−0.0988 − 0.995i)2-s + (−0.980 + 0.196i)4-s + 0.656i·5-s − 0.377i·7-s + (0.292 + 0.956i)8-s + (0.653 − 0.0649i)10-s − 1.15·11-s − 1.42·13-s + (−0.376 + 0.0373i)14-s + (0.922 − 0.385i)16-s − 0.279i·17-s − 1.68i·19-s + (−0.129 − 0.644i)20-s + (0.114 + 1.15i)22-s − 0.916·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.914 + 0.405i$
Analytic conductor: \(78.7210\)
Root analytic conductor: \(8.87248\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :7/2),\ 0.914 + 0.405i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.9280915190\)
\(L(\frac12)\) \(\approx\) \(0.9280915190\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.11 + 11.2i)T \)
3 \( 1 \)
7 \( 1 + 343iT \)
good5 \( 1 - 183. iT - 7.81e4T^{2} \)
11 \( 1 + 5.11e3T + 1.94e7T^{2} \)
13 \( 1 + 1.12e4T + 6.27e7T^{2} \)
17 \( 1 + 5.65e3iT - 4.10e8T^{2} \)
19 \( 1 + 5.02e4iT - 8.93e8T^{2} \)
23 \( 1 + 5.34e4T + 3.40e9T^{2} \)
29 \( 1 + 1.39e5iT - 1.72e10T^{2} \)
31 \( 1 - 1.50e5iT - 2.75e10T^{2} \)
37 \( 1 + 4.52e5T + 9.49e10T^{2} \)
41 \( 1 - 2.95e5iT - 1.94e11T^{2} \)
43 \( 1 - 5.30e5iT - 2.71e11T^{2} \)
47 \( 1 - 1.35e6T + 5.06e11T^{2} \)
53 \( 1 - 1.62e6iT - 1.17e12T^{2} \)
59 \( 1 - 8.02e5T + 2.48e12T^{2} \)
61 \( 1 - 2.49e6T + 3.14e12T^{2} \)
67 \( 1 - 2.94e6iT - 6.06e12T^{2} \)
71 \( 1 - 3.55e6T + 9.09e12T^{2} \)
73 \( 1 + 3.08e6T + 1.10e13T^{2} \)
79 \( 1 + 7.02e6iT - 1.92e13T^{2} \)
83 \( 1 - 6.17e6T + 2.71e13T^{2} \)
89 \( 1 + 8.07e5iT - 4.42e13T^{2} \)
97 \( 1 + 8.56e6T + 8.07e13T^{2} \)
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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

−10.61277438242458684811178069078, −10.10273249433235994113208328998, −9.062929027985996284584098004522, −7.83029941889500217005059813015, −6.99547530669585390091635099515, −5.29483120916013467081283063110, −4.42340808370422173339664348935, −2.92381000176440470383418426336, −2.34675832153095221781959276770, −0.59496154255102848773093231889, 0.38443305499643522849818365545, 2.05926226250052971304869960167, 3.81792216883083550815448210446, 5.13694266008677916060999934056, 5.58612767012776963292360183568, 7.02352470395693179450444576481, 7.956561549918513436724660580082, 8.660043815342858991648748323758, 9.794207315853063755985021386419, 10.46161140719235076219321983595

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