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Properties

Label 2-336-7.2-c5-0-32
Degree 22
Conductor 336336
Sign −0.963+0.267i-0.963 + 0.267i
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 + (−11.2 + 19.4i)5-s + (96.4 − 86.5i)7-s + (−40.5 + 70.1i)9-s + (170. + 294. i)11-s − 728.·13-s + 202.·15-s + (−404. − 701. i)17-s + (513. − 888. i)19-s + (−1.10e3 − 362. i)21-s + (711. − 1.23e3i)23-s + (1.30e3 + 2.26e3i)25-s + 729·27-s + 5.21e3·29-s + (−3.51e3 − 6.09e3i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.201 + 0.348i)5-s + (0.744 − 0.667i)7-s + (−0.166 + 0.288i)9-s + (0.424 + 0.734i)11-s − 1.19·13-s + 0.232·15-s + (−0.339 − 0.588i)17-s + (0.326 − 0.564i)19-s + (−0.548 − 0.179i)21-s + (0.280 − 0.485i)23-s + (0.419 + 0.725i)25-s + 0.192·27-s + 1.15·29-s + (−0.657 − 1.13i)31-s + ⋯

Functional equation

Λ(s)=(336s/2 ΓC(s) L(s)=((−0.963+0.267i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.267i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(336s/2 ΓC(s+5/2) L(s)=((−0.963+0.267i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 336336    =    24⋅3⋅72^{4} \cdot 3 \cdot 7
Sign: −0.963+0.267i-0.963 + 0.267i
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), −0.963+0.267i)(2,\ 336,\ (\ :5/2),\ -0.963 + 0.267i)

Particular Values

L(3)L(3) ≈\approx 0.65309769920.6530976992
L(12)L(\frac12) ≈\approx 0.65309769920.6530976992
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+(−96.4+86.5i)T 1 + (-96.4 + 86.5i)T
good5 1+(11.2−19.4i)T+(−1.56e3−2.70e3i)T2 1 + (11.2 - 19.4i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(−170.−294.i)T+(−8.05e4+1.39e5i)T2 1 + (-170. - 294. i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1+728.T+3.71e5T2 1 + 728.T + 3.71e5T^{2}
17 1+(404.+701.i)T+(−7.09e5+1.22e6i)T2 1 + (404. + 701. i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(−513.+888.i)T+(−1.23e6−2.14e6i)T2 1 + (-513. + 888. i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(−711.+1.23e3i)T+(−3.21e6−5.57e6i)T2 1 + (-711. + 1.23e3i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1−5.21e3T+2.05e7T2 1 - 5.21e3T + 2.05e7T^{2}
31 1+(3.51e3+6.09e3i)T+(−1.43e7+2.47e7i)T2 1 + (3.51e3 + 6.09e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(6.39e3−1.10e4i)T+(−3.46e7−6.00e7i)T2 1 + (6.39e3 - 1.10e4i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1−1.17e3T+1.15e8T2 1 - 1.17e3T + 1.15e8T^{2}
43 1+3.66e3T+1.47e8T2 1 + 3.66e3T + 1.47e8T^{2}
47 1+(−4.65e3+8.06e3i)T+(−1.14e8−1.98e8i)T2 1 + (-4.65e3 + 8.06e3i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(1.78e4+3.08e4i)T+(−2.09e8+3.62e8i)T2 1 + (1.78e4 + 3.08e4i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(1.51e4+2.63e4i)T+(−3.57e8+6.19e8i)T2 1 + (1.51e4 + 2.63e4i)T + (-3.57e8 + 6.19e8i)T^{2}
61 1+(1.60e4−2.78e4i)T+(−4.22e8−7.31e8i)T2 1 + (1.60e4 - 2.78e4i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(−1.06e4−1.84e4i)T+(−6.75e8+1.16e9i)T2 1 + (-1.06e4 - 1.84e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1+6.11e4T+1.80e9T2 1 + 6.11e4T + 1.80e9T^{2}
73 1+(2.06e4+3.57e4i)T+(−1.03e9+1.79e9i)T2 1 + (2.06e4 + 3.57e4i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(1.75e4−3.03e4i)T+(−1.53e9−2.66e9i)T2 1 + (1.75e4 - 3.03e4i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1+8.61e4T+3.93e9T2 1 + 8.61e4T + 3.93e9T^{2}
89 1+(3.89e4−6.75e4i)T+(−2.79e9−4.83e9i)T2 1 + (3.89e4 - 6.75e4i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1+1.61e5T+8.58e9T2 1 + 1.61e5T + 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

−10.39501857978055677288549894001, −9.465248455170271710804489528777, −8.204629951384130880292115367911, −7.17049658544612904984236278780, −6.84376097425132941238562797928, −5.16161245833028639427157226494, −4.42445899417816620148550033226, −2.80425232209686859272645099190, −1.53371413056489809555284023335, −0.17772440358612500032284825227, 1.39998275367882146143241893666, 2.91126666527624320595977150986, 4.30010729937356857222730498208, 5.16599322793603760425774586350, 6.08977406802697346458534352411, 7.43405440489018828499527026641, 8.542116378183797038172406321602, 9.140383908086825407899642182783, 10.33453607095052499775651833984, 11.12371789613013926100231121383

Graph of the ZZ-function along the critical line