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Properties

Label 2-416-13.3-c3-0-13
Degree 22
Conductor 416416
Sign 0.522+0.852i0.522 + 0.852i
Analytic cond. 24.544724.5447
Root an. cond. 4.954274.95427
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.25 − 7.37i)3-s − 9.83·5-s + (−12.0 + 20.8i)7-s + (−22.7 + 39.4i)9-s + (14.2 + 24.5i)11-s + (−13 − 45.0i)13-s + (41.8 + 72.4i)15-s + (16.5 − 28.7i)17-s + (3.54 − 6.13i)19-s + 205.·21-s + (98.6 + 170. i)23-s − 28.3·25-s + 157.·27-s + (−11.3 − 19.6i)29-s − 48.3·31-s + ⋯
L(s)  = 1  + (−0.819 − 1.41i)3-s − 0.879·5-s + (−0.651 + 1.12i)7-s + (−0.842 + 1.45i)9-s + (0.389 + 0.674i)11-s + (−0.277 − 0.960i)13-s + (0.720 + 1.24i)15-s + (0.236 − 0.409i)17-s + (0.0427 − 0.0740i)19-s + 2.13·21-s + (0.894 + 1.54i)23-s − 0.226·25-s + 1.12·27-s + (−0.0725 − 0.125i)29-s − 0.280·31-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((0.522+0.852i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+3/2) L(s)=((0.522+0.852i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 0.522+0.852i0.522 + 0.852i
Analytic conductor: 24.544724.5447
Root analytic conductor: 4.954274.95427
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ416(289,⋅)\chi_{416} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :3/2), 0.522+0.852i)(2,\ 416,\ (\ :3/2),\ 0.522 + 0.852i)

Particular Values

L(2)L(2) ≈\approx 0.77859388020.7785938802
L(12)L(\frac12) ≈\approx 0.77859388020.7785938802
L(52)L(\frac{5}{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
13 1+(13+45.0i)T 1 + (13 + 45.0i)T
good3 1+(4.25+7.37i)T+(−13.5+23.3i)T2 1 + (4.25 + 7.37i)T + (-13.5 + 23.3i)T^{2}
5 1+9.83T+125T2 1 + 9.83T + 125T^{2}
7 1+(12.0−20.8i)T+(−171.5−297.i)T2 1 + (12.0 - 20.8i)T + (-171.5 - 297. i)T^{2}
11 1+(−14.2−24.5i)T+(−665.5+1.15e3i)T2 1 + (-14.2 - 24.5i)T + (-665.5 + 1.15e3i)T^{2}
17 1+(−16.5+28.7i)T+(−2.45e3−4.25e3i)T2 1 + (-16.5 + 28.7i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(−3.54+6.13i)T+(−3.42e3−5.94e3i)T2 1 + (-3.54 + 6.13i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(−98.6−170.i)T+(−6.08e3+1.05e4i)T2 1 + (-98.6 - 170. i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+(11.3+19.6i)T+(−1.21e4+2.11e4i)T2 1 + (11.3 + 19.6i)T + (-1.21e4 + 2.11e4i)T^{2}
31 1+48.3T+2.97e4T2 1 + 48.3T + 2.97e4T^{2}
37 1+(7.58+13.1i)T+(−2.53e4+4.38e4i)T2 1 + (7.58 + 13.1i)T + (-2.53e4 + 4.38e4i)T^{2}
41 1+(237.+411.i)T+(−3.44e4+5.96e4i)T2 1 + (237. + 411. i)T + (-3.44e4 + 5.96e4i)T^{2}
43 1+(−120.+208.i)T+(−3.97e4−6.88e4i)T2 1 + (-120. + 208. i)T + (-3.97e4 - 6.88e4i)T^{2}
47 1+214.T+1.03e5T2 1 + 214.T + 1.03e5T^{2}
53 1−719.T+1.48e5T2 1 - 719.T + 1.48e5T^{2}
59 1+(−71.6+124.i)T+(−1.02e5−1.77e5i)T2 1 + (-71.6 + 124. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(258.−448.i)T+(−1.13e5−1.96e5i)T2 1 + (258. - 448. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(308.+534.i)T+(−1.50e5+2.60e5i)T2 1 + (308. + 534. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1+(131.−228.i)T+(−1.78e5−3.09e5i)T2 1 + (131. - 228. i)T + (-1.78e5 - 3.09e5i)T^{2}
73 1−1.05e3T+3.89e5T2 1 - 1.05e3T + 3.89e5T^{2}
79 1−1.25e3T+4.93e5T2 1 - 1.25e3T + 4.93e5T^{2}
83 1+224.T+5.71e5T2 1 + 224.T + 5.71e5T^{2}
89 1+(635.+1.10e3i)T+(−3.52e5+6.10e5i)T2 1 + (635. + 1.10e3i)T + (-3.52e5 + 6.10e5i)T^{2}
97 1+(398.−689.i)T+(−4.56e5−7.90e5i)T2 1 + (398. - 689. i)T + (-4.56e5 - 7.90e5i)T^{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

−11.01221484907962099997463386450, −9.707210359031379128481366337696, −8.641527818305051535226346715958, −7.48776463648528268705690500215, −7.10410522347820295739423686204, −5.88124959958777562870168700374, −5.22096385699861503563605393229, −3.41176518770997373352972770374, −2.06548669051693363237331724743, −0.54587146524662132648326596620, 0.61802569025816099976861180147, 3.37144041456580217387998442741, 4.08804058762572754142172369288, 4.80176837604238711604571138242, 6.19965827432163144349277213051, 6.99962333911680142738080591749, 8.338897299498019244999422892129, 9.394569378592593233637952054818, 10.16436523937633626800816881218, 10.94147054069170432788316955122

Graph of the ZZ-function along the critical line