Properties

Label 2-416-13.4-c1-0-12
Degree 22
Conductor 416416
Sign 0.265+0.964i-0.265 + 0.964i
Analytic cond. 3.321773.32177
Root an. cond. 1.822571.82257
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 1.5i)3-s − 3.46i·5-s + (2.59 − 1.5i)7-s + (−4.33 − 2.5i)11-s + (−1 + 3.46i)13-s + (−5.19 − 2.99i)15-s + (3.5 + 6.06i)17-s + (−4.33 + 2.5i)19-s − 5.19i·21-s + (2.59 − 4.5i)23-s − 6.99·25-s + 5.19·27-s + (2.5 − 4.33i)29-s − 2i·31-s + (−7.5 + 4.33i)33-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)3-s − 1.54i·5-s + (0.981 − 0.566i)7-s + (−1.30 − 0.753i)11-s + (−0.277 + 0.960i)13-s + (−1.34 − 0.774i)15-s + (0.848 + 1.47i)17-s + (−0.993 + 0.573i)19-s − 1.13i·21-s + (0.541 − 0.938i)23-s − 1.39·25-s + 1.00·27-s + (0.464 − 0.804i)29-s − 0.359i·31-s + (−1.30 + 0.753i)33-s + ⋯

Functional equation

Λ(s)=(416s/2ΓC(s)L(s)=((0.265+0.964i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2ΓC(s+1/2)L(s)=((0.265+0.964i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25132^{5} \cdot 13
Sign: 0.265+0.964i-0.265 + 0.964i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(225,)\chi_{416} (225, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), 0.265+0.964i)(2,\ 416,\ (\ :1/2),\ -0.265 + 0.964i)

Particular Values

L(1)L(1) \approx 1.004321.31759i1.00432 - 1.31759i
L(12)L(\frac12) \approx 1.004321.31759i1.00432 - 1.31759i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
13 1+(13.46i)T 1 + (1 - 3.46i)T
good3 1+(0.866+1.5i)T+(1.52.59i)T2 1 + (-0.866 + 1.5i)T + (-1.5 - 2.59i)T^{2}
5 1+3.46iT5T2 1 + 3.46iT - 5T^{2}
7 1+(2.59+1.5i)T+(3.56.06i)T2 1 + (-2.59 + 1.5i)T + (3.5 - 6.06i)T^{2}
11 1+(4.33+2.5i)T+(5.5+9.52i)T2 1 + (4.33 + 2.5i)T + (5.5 + 9.52i)T^{2}
17 1+(3.56.06i)T+(8.5+14.7i)T2 1 + (-3.5 - 6.06i)T + (-8.5 + 14.7i)T^{2}
19 1+(4.332.5i)T+(9.516.4i)T2 1 + (4.33 - 2.5i)T + (9.5 - 16.4i)T^{2}
23 1+(2.59+4.5i)T+(11.519.9i)T2 1 + (-2.59 + 4.5i)T + (-11.5 - 19.9i)T^{2}
29 1+(2.5+4.33i)T+(14.525.1i)T2 1 + (-2.5 + 4.33i)T + (-14.5 - 25.1i)T^{2}
31 1+2iT31T2 1 + 2iT - 31T^{2}
37 1+(4.52.59i)T+(18.5+32.0i)T2 1 + (-4.5 - 2.59i)T + (18.5 + 32.0i)T^{2}
41 1+(1.5+0.866i)T+(20.5+35.5i)T2 1 + (1.5 + 0.866i)T + (20.5 + 35.5i)T^{2}
43 1+(2.594.5i)T+(21.5+37.2i)T2 1 + (-2.59 - 4.5i)T + (-21.5 + 37.2i)T^{2}
47 1+4iT47T2 1 + 4iT - 47T^{2}
53 14T+53T2 1 - 4T + 53T^{2}
59 1+(6.06+3.5i)T+(29.551.0i)T2 1 + (-6.06 + 3.5i)T + (29.5 - 51.0i)T^{2}
61 1+(1.5+2.59i)T+(30.5+52.8i)T2 1 + (1.5 + 2.59i)T + (-30.5 + 52.8i)T^{2}
67 1+(2.59+1.5i)T+(33.5+58.0i)T2 1 + (2.59 + 1.5i)T + (33.5 + 58.0i)T^{2}
71 1+(6.063.5i)T+(35.561.4i)T2 1 + (6.06 - 3.5i)T + (35.5 - 61.4i)T^{2}
73 1+3.46iT73T2 1 + 3.46iT - 73T^{2}
79 1+3.46T+79T2 1 + 3.46T + 79T^{2}
83 114iT83T2 1 - 14iT - 83T^{2}
89 1+(1.50.866i)T+(44.5+77.0i)T2 1 + (-1.5 - 0.866i)T + (44.5 + 77.0i)T^{2}
97 1+(7.5+4.33i)T+(48.584.0i)T2 1 + (-7.5 + 4.33i)T + (48.5 - 84.0i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−10.94908538175983714923386239358, −10.08894683545246689525724414725, −8.592667418902307145411061986442, −8.252334559306368302886744114880, −7.65184186003859798424435421020, −6.18570872055557231223196832515, −4.99253488121029589828855222896, −4.19300724258586834100372526043, −2.20001653520271263739376382071, −1.11471873287324465725549016299, 2.54894164571844234500184399434, 3.13280708837345839580850573153, 4.69551890972061568647975851003, 5.51169308262128402487868568562, 7.09480649625657389865806807161, 7.67431865783985467018022019400, 8.830804606247835746766407911564, 9.922392955102943269953430468258, 10.45289529232732521623226162730, 11.18730186715297796826659150158

Graph of the ZZ-function along the critical line