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Properties

Label 2-252-7.4-c11-0-19
Degree $2$
Conductor $252$
Sign $0.793 + 0.609i$
Analytic cond. $193.622$
Root an. cond. $13.9148$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.96e3 + 1.03e4i)5-s + (−1.95e4 − 3.99e4i)7-s + (−2.25e5 + 3.91e5i)11-s − 2.01e6·13-s + (1.36e6 − 2.36e6i)17-s + (3.24e6 + 5.62e6i)19-s + (−1.08e7 − 1.88e7i)23-s + (−4.68e7 + 8.11e7i)25-s − 1.78e8·29-s + (3.08e7 − 5.33e7i)31-s + (2.96e8 − 4.40e8i)35-s + (1.65e8 + 2.86e8i)37-s + 1.76e8·41-s + 1.54e9·43-s + (−1.20e9 − 2.09e9i)47-s + ⋯
L(s)  = 1  + (0.854 + 1.47i)5-s + (−0.439 − 0.898i)7-s + (−0.422 + 0.732i)11-s − 1.50·13-s + (0.233 − 0.403i)17-s + (0.300 + 0.520i)19-s + (−0.352 − 0.609i)23-s + (−0.959 + 1.66i)25-s − 1.61·29-s + (0.193 − 0.334i)31-s + (0.952 − 1.41i)35-s + (0.392 + 0.679i)37-s + 0.238·41-s + 1.60·43-s + (−0.768 − 1.33i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.609i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.793 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.793 + 0.609i$
Analytic conductor: \(193.622\)
Root analytic conductor: \(13.9148\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :11/2),\ 0.793 + 0.609i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.362235622\)
\(L(\frac12)\) \(\approx\) \(1.362235622\)
\(L(\frac{13}{2})\) 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 \( 1 + (1.95e4 + 3.99e4i)T \)
good5 \( 1 + (-5.96e3 - 1.03e4i)T + (-2.44e7 + 4.22e7i)T^{2} \)
11 \( 1 + (2.25e5 - 3.91e5i)T + (-1.42e11 - 2.47e11i)T^{2} \)
13 \( 1 + 2.01e6T + 1.79e12T^{2} \)
17 \( 1 + (-1.36e6 + 2.36e6i)T + (-1.71e13 - 2.96e13i)T^{2} \)
19 \( 1 + (-3.24e6 - 5.62e6i)T + (-5.82e13 + 1.00e14i)T^{2} \)
23 \( 1 + (1.08e7 + 1.88e7i)T + (-4.76e14 + 8.25e14i)T^{2} \)
29 \( 1 + 1.78e8T + 1.22e16T^{2} \)
31 \( 1 + (-3.08e7 + 5.33e7i)T + (-1.27e16 - 2.20e16i)T^{2} \)
37 \( 1 + (-1.65e8 - 2.86e8i)T + (-8.89e16 + 1.54e17i)T^{2} \)
41 \( 1 - 1.76e8T + 5.50e17T^{2} \)
43 \( 1 - 1.54e9T + 9.29e17T^{2} \)
47 \( 1 + (1.20e9 + 2.09e9i)T + (-1.23e18 + 2.14e18i)T^{2} \)
53 \( 1 + (-2.67e9 + 4.63e9i)T + (-4.63e18 - 8.02e18i)T^{2} \)
59 \( 1 + (4.67e7 - 8.09e7i)T + (-1.50e19 - 2.61e19i)T^{2} \)
61 \( 1 + (-2.30e9 - 3.98e9i)T + (-2.17e19 + 3.76e19i)T^{2} \)
67 \( 1 + (-6.33e9 + 1.09e10i)T + (-6.10e19 - 1.05e20i)T^{2} \)
71 \( 1 + 1.03e10T + 2.31e20T^{2} \)
73 \( 1 + (-2.83e9 + 4.91e9i)T + (-1.56e20 - 2.71e20i)T^{2} \)
79 \( 1 + (2.64e10 + 4.57e10i)T + (-3.73e20 + 6.47e20i)T^{2} \)
83 \( 1 - 8.41e9T + 1.28e21T^{2} \)
89 \( 1 + (-3.71e10 - 6.42e10i)T + (-1.38e21 + 2.40e21i)T^{2} \)
97 \( 1 + 6.08e10T + 7.15e21T^{2} \)
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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

−9.996609184308191600178378797131, −9.571790248133191727613300994165, −7.60762996327193982178305212833, −7.18351918663860838859611572506, −6.24813204112358462687853865383, −5.10861625445399750628896124599, −3.77881672251966291955674468180, −2.68468945486402118857650931721, −1.97328994125196860790418036789, −0.30344351510864707269217108471, 0.73992858267278411582435966005, 1.92201064331424254929939243993, 2.82276584688668749534701755966, 4.40054668691273127593350825460, 5.54567752104000110981980905688, 5.75543689343050221628066594052, 7.42100671655815407369876828795, 8.507676325470468044660482154713, 9.362286954866675783521522533420, 9.773010080639761954515180047466

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