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Properties

Label 2-336-7.2-c5-0-6
Degree 22
Conductor 336336
Sign −0.702+0.711i-0.702 + 0.711i
Analytic cond. 53.888953.8889
Root an. cond. 7.340917.34091
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.5 + 7.79i)3-s + (−32.7 + 56.7i)5-s + (−40.6 + 123. i)7-s + (−40.5 + 70.1i)9-s + (−122. − 212. i)11-s + 434.·13-s − 590.·15-s + (551. + 954. i)17-s + (−1.43e3 + 2.49e3i)19-s + (−1.14e3 + 237. i)21-s + (−2.11e3 + 3.66e3i)23-s + (−587. − 1.01e3i)25-s − 729·27-s + 4.96e3·29-s + (−4.39e3 − 7.60e3i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.586 + 1.01i)5-s + (−0.313 + 0.949i)7-s + (−0.166 + 0.288i)9-s + (−0.305 − 0.529i)11-s + 0.713·13-s − 0.677·15-s + (0.462 + 0.801i)17-s + (−0.914 + 1.58i)19-s + (−0.565 + 0.117i)21-s + (−0.833 + 1.44i)23-s + (−0.187 − 0.325i)25-s − 0.192·27-s + 1.09·29-s + (−0.820 − 1.42i)31-s + ⋯

Functional equation

Λ(s)=(336s/2 ΓC(s) L(s)=((−0.702+0.711i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(336s/2 ΓC(s+5/2) L(s)=((−0.702+0.711i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 336336    =    24⋅3⋅72^{4} \cdot 3 \cdot 7
Sign: −0.702+0.711i-0.702 + 0.711i
Analytic conductor: 53.888953.8889
Root analytic conductor: 7.340917.34091
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ336(289,⋅)\chi_{336} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 336, ( :5/2), −0.702+0.711i)(2,\ 336,\ (\ :5/2),\ -0.702 + 0.711i)

Particular Values

L(3)L(3) ≈\approx 0.89870968710.8987096871
L(12)L(\frac12) ≈\approx 0.89870968710.8987096871
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(−4.5−7.79i)T 1 + (-4.5 - 7.79i)T
7 1+(40.6−123.i)T 1 + (40.6 - 123. i)T
good5 1+(32.7−56.7i)T+(−1.56e3−2.70e3i)T2 1 + (32.7 - 56.7i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(122.+212.i)T+(−8.05e4+1.39e5i)T2 1 + (122. + 212. i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1−434.T+3.71e5T2 1 - 434.T + 3.71e5T^{2}
17 1+(−551.−954.i)T+(−7.09e5+1.22e6i)T2 1 + (-551. - 954. i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(1.43e3−2.49e3i)T+(−1.23e6−2.14e6i)T2 1 + (1.43e3 - 2.49e3i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(2.11e3−3.66e3i)T+(−3.21e6−5.57e6i)T2 1 + (2.11e3 - 3.66e3i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1−4.96e3T+2.05e7T2 1 - 4.96e3T + 2.05e7T^{2}
31 1+(4.39e3+7.60e3i)T+(−1.43e7+2.47e7i)T2 1 + (4.39e3 + 7.60e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(−1.22e3+2.11e3i)T+(−3.46e7−6.00e7i)T2 1 + (-1.22e3 + 2.11e3i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1+3.66e3T+1.15e8T2 1 + 3.66e3T + 1.15e8T^{2}
43 1−7.19e3T+1.47e8T2 1 - 7.19e3T + 1.47e8T^{2}
47 1+(−1.63e3+2.83e3i)T+(−1.14e8−1.98e8i)T2 1 + (-1.63e3 + 2.83e3i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(−1.51e3−2.61e3i)T+(−2.09e8+3.62e8i)T2 1 + (-1.51e3 - 2.61e3i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(2.57e4+4.45e4i)T+(−3.57e8+6.19e8i)T2 1 + (2.57e4 + 4.45e4i)T + (-3.57e8 + 6.19e8i)T^{2}
61 1+(−6.65e3+1.15e4i)T+(−4.22e8−7.31e8i)T2 1 + (-6.65e3 + 1.15e4i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(−1.54e4−2.67e4i)T+(−6.75e8+1.16e9i)T2 1 + (-1.54e4 - 2.67e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1−4.18e4T+1.80e9T2 1 - 4.18e4T + 1.80e9T^{2}
73 1+(1.73e4+2.99e4i)T+(−1.03e9+1.79e9i)T2 1 + (1.73e4 + 2.99e4i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(−3.87e4+6.71e4i)T+(−1.53e9−2.66e9i)T2 1 + (-3.87e4 + 6.71e4i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1+1.00e5T+3.93e9T2 1 + 1.00e5T + 3.93e9T^{2}
89 1+(2.03e4−3.53e4i)T+(−2.79e9−4.83e9i)T2 1 + (2.03e4 - 3.53e4i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1−1.40e5T+8.58e9T2 1 - 1.40e5T + 8.58e9T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−11.17505037092717498926217695595, −10.43440422303049655290567575630, −9.531390322965518325499882400064, −8.354231335894596811151225647118, −7.79573323556952346675004793094, −6.27022547123651092358562959191, −5.66524850394053160899497326323, −3.87591368335653290594814474029, −3.30871963207071175035395504726, −1.98325521235115412132231605525, 0.24671663315313065664394785876, 1.10015129013693860208927195640, 2.72119513200681948576377328553, 4.13742844870521718234176784084, 4.88642637732233273015382290831, 6.49971569999435894891546937498, 7.28171789407181508001249805906, 8.326608439416653128220680522322, 8.938535003942496739267139182125, 10.17280402876628905283259666346

Graph of the ZZ-function along the critical line