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Properties

Label 2-252-21.17-c11-0-15
Degree $2$
Conductor $252$
Sign $0.938 - 0.344i$
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  + (5.67e3 + 9.82e3i)5-s + (4.40e4 − 5.90e3i)7-s + (−8.47e5 − 4.89e5i)11-s + 5.43e5i·13-s + (1.36e6 − 2.36e6i)17-s + (−3.62e6 + 2.09e6i)19-s + (−6.66e6 + 3.84e6i)23-s + (−3.99e7 + 6.91e7i)25-s − 2.06e8i·29-s + (4.21e7 + 2.43e7i)31-s + (3.08e8 + 3.99e8i)35-s + (−2.72e8 − 4.71e8i)37-s + 1.42e9·41-s + 1.26e9·43-s + (5.06e8 + 8.77e8i)47-s + ⋯
L(s)  = 1  + (0.811 + 1.40i)5-s + (0.991 − 0.132i)7-s + (−1.58 − 0.916i)11-s + 0.406i·13-s + (0.232 − 0.403i)17-s + (−0.335 + 0.193i)19-s + (−0.215 + 0.124i)23-s + (−0.817 + 1.41i)25-s − 1.87i·29-s + (0.264 + 0.152i)31-s + (0.991 + 1.28i)35-s + (−0.646 − 1.11i)37-s + 1.92·41-s + 1.30·43-s + (0.322 + 0.557i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.938 - 0.344i$
Analytic conductor: \(193.622\)
Root analytic conductor: \(13.9148\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :11/2),\ 0.938 - 0.344i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.801392372\)
\(L(\frac12)\) \(\approx\) \(2.801392372\)
\(L(\frac{13}{2})\) 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 \( 1 + (-4.40e4 + 5.90e3i)T \)
good5 \( 1 + (-5.67e3 - 9.82e3i)T + (-2.44e7 + 4.22e7i)T^{2} \)
11 \( 1 + (8.47e5 + 4.89e5i)T + (1.42e11 + 2.47e11i)T^{2} \)
13 \( 1 - 5.43e5iT - 1.79e12T^{2} \)
17 \( 1 + (-1.36e6 + 2.36e6i)T + (-1.71e13 - 2.96e13i)T^{2} \)
19 \( 1 + (3.62e6 - 2.09e6i)T + (5.82e13 - 1.00e14i)T^{2} \)
23 \( 1 + (6.66e6 - 3.84e6i)T + (4.76e14 - 8.25e14i)T^{2} \)
29 \( 1 + 2.06e8iT - 1.22e16T^{2} \)
31 \( 1 + (-4.21e7 - 2.43e7i)T + (1.27e16 + 2.20e16i)T^{2} \)
37 \( 1 + (2.72e8 + 4.71e8i)T + (-8.89e16 + 1.54e17i)T^{2} \)
41 \( 1 - 1.42e9T + 5.50e17T^{2} \)
43 \( 1 - 1.26e9T + 9.29e17T^{2} \)
47 \( 1 + (-5.06e8 - 8.77e8i)T + (-1.23e18 + 2.14e18i)T^{2} \)
53 \( 1 + (-3.80e9 - 2.19e9i)T + (4.63e18 + 8.02e18i)T^{2} \)
59 \( 1 + (-3.96e9 + 6.86e9i)T + (-1.50e19 - 2.61e19i)T^{2} \)
61 \( 1 + (1.92e9 - 1.11e9i)T + (2.17e19 - 3.76e19i)T^{2} \)
67 \( 1 + (1.39e9 - 2.41e9i)T + (-6.10e19 - 1.05e20i)T^{2} \)
71 \( 1 - 1.07e10iT - 2.31e20T^{2} \)
73 \( 1 + (1.48e9 + 8.58e8i)T + (1.56e20 + 2.71e20i)T^{2} \)
79 \( 1 + (1.58e10 + 2.75e10i)T + (-3.73e20 + 6.47e20i)T^{2} \)
83 \( 1 - 3.27e10T + 1.28e21T^{2} \)
89 \( 1 + (-2.15e10 - 3.72e10i)T + (-1.38e21 + 2.40e21i)T^{2} \)
97 \( 1 - 1.40e11iT - 7.15e21T^{2} \)
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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

−10.50042977840383320298295305603, −9.329621152213532852489295523674, −8.004681885141663011171748954346, −7.38183418743890236960710102073, −6.08644162154360549336891933334, −5.46119544214386819826679488796, −4.05845572731010629638689525471, −2.64936008344325950161575078862, −2.22418773129495835886086255089, −0.66550512928724035148587828119, 0.74788683553282744250356352826, 1.69462969700084448199460371627, 2.54276717265565196428770087427, 4.39326032803238815175525572120, 5.13452977229713869745798120700, 5.69548957296790887061633047117, 7.36359707533030824394010334500, 8.267328934926850087807053686985, 8.962642277178804024520573324049, 10.09877365872857386593586303115

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