Properties

Label 2-252-7.4-c11-0-16
Degree $2$
Conductor $252$
Sign $0.964 - 0.264i$
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  + (1.33e3 + 2.30e3i)5-s + (4.36e4 + 8.49e3i)7-s + (−2.62e5 + 4.55e5i)11-s − 1.10e6·13-s + (2.22e5 − 3.84e5i)17-s + (−1.58e6 − 2.74e6i)19-s + (−2.38e7 − 4.13e7i)23-s + (2.08e7 − 3.61e7i)25-s + 1.65e8·29-s + (−8.05e7 + 1.39e8i)31-s + (3.85e7 + 1.12e8i)35-s + (−1.73e8 − 3.00e8i)37-s + 1.07e9·41-s + 1.10e9·43-s + (4.49e7 + 7.77e7i)47-s + ⋯
L(s)  = 1  + (0.190 + 0.330i)5-s + (0.981 + 0.191i)7-s + (−0.492 + 0.852i)11-s − 0.823·13-s + (0.0379 − 0.0656i)17-s + (−0.146 − 0.254i)19-s + (−0.773 − 1.34i)23-s + (0.427 − 0.740i)25-s + 1.49·29-s + (−0.505 + 0.875i)31-s + (0.124 + 0.360i)35-s + (−0.410 − 0.711i)37-s + 1.44·41-s + 1.14·43-s + (0.0285 + 0.0494i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.264i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.964 - 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.964 - 0.264i$
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: \((2,\ 252,\ (\ :11/2),\ 0.964 - 0.264i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.428157201\)
\(L(\frac12)\) \(\approx\) \(2.428157201\)
\(L(\frac{13}{2})\) 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 \( 1 + (-4.36e4 - 8.49e3i)T \)
good5 \( 1 + (-1.33e3 - 2.30e3i)T + (-2.44e7 + 4.22e7i)T^{2} \)
11 \( 1 + (2.62e5 - 4.55e5i)T + (-1.42e11 - 2.47e11i)T^{2} \)
13 \( 1 + 1.10e6T + 1.79e12T^{2} \)
17 \( 1 + (-2.22e5 + 3.84e5i)T + (-1.71e13 - 2.96e13i)T^{2} \)
19 \( 1 + (1.58e6 + 2.74e6i)T + (-5.82e13 + 1.00e14i)T^{2} \)
23 \( 1 + (2.38e7 + 4.13e7i)T + (-4.76e14 + 8.25e14i)T^{2} \)
29 \( 1 - 1.65e8T + 1.22e16T^{2} \)
31 \( 1 + (8.05e7 - 1.39e8i)T + (-1.27e16 - 2.20e16i)T^{2} \)
37 \( 1 + (1.73e8 + 3.00e8i)T + (-8.89e16 + 1.54e17i)T^{2} \)
41 \( 1 - 1.07e9T + 5.50e17T^{2} \)
43 \( 1 - 1.10e9T + 9.29e17T^{2} \)
47 \( 1 + (-4.49e7 - 7.77e7i)T + (-1.23e18 + 2.14e18i)T^{2} \)
53 \( 1 + (1.97e8 - 3.42e8i)T + (-4.63e18 - 8.02e18i)T^{2} \)
59 \( 1 + (3.65e9 - 6.32e9i)T + (-1.50e19 - 2.61e19i)T^{2} \)
61 \( 1 + (6.05e9 + 1.04e10i)T + (-2.17e19 + 3.76e19i)T^{2} \)
67 \( 1 + (-8.63e9 + 1.49e10i)T + (-6.10e19 - 1.05e20i)T^{2} \)
71 \( 1 - 1.31e10T + 2.31e20T^{2} \)
73 \( 1 + (7.39e9 - 1.28e10i)T + (-1.56e20 - 2.71e20i)T^{2} \)
79 \( 1 + (1.45e10 + 2.51e10i)T + (-3.73e20 + 6.47e20i)T^{2} \)
83 \( 1 + 2.43e9T + 1.28e21T^{2} \)
89 \( 1 + (-4.46e10 - 7.73e10i)T + (-1.38e21 + 2.40e21i)T^{2} \)
97 \( 1 - 6.63e10T + 7.15e21T^{2} \)
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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

−10.34115707998617110572060386270, −9.187626930934932557588770374013, −8.145504300783828793318440676195, −7.31990932757842983325582698990, −6.26880100869699539910043183514, −4.98741914269369205671715759508, −4.38598306761671867860699867819, −2.66503789779317326907691068259, −2.05766489806816605823199505451, −0.65263963380296872451600853685, 0.66579353185863972853120481445, 1.66191399229931823655731984680, 2.80700407580342462358258891259, 4.12793397151430034353632144130, 5.14197302344537922759087301800, 5.92449904542143900171839539670, 7.39553144509912631685194372303, 8.071633503419484379937240693857, 9.062128623177926129350447519699, 10.10035748784713784947047474132

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