↑

Properties

Label 2-416-13.10-c1-0-10
Degree 22
Conductor 416416
Sign 0.265+0.964i0.265 + 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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.619 − 1.07i)3-s − 2i·5-s + (4.00 + 2.31i)7-s + (0.732 − 1.26i)9-s + (−1.85 + 1.07i)11-s + (−1 − 3.46i)13-s + (−2.14 + 1.23i)15-s + (0.232 − 0.401i)17-s + (4.00 + 2.31i)19-s − 5.73i·21-s + (−2.76 − 4.79i)23-s + 25-s − 5.53·27-s + (−1.5 − 2.59i)29-s − 9.25i·31-s + ⋯
L(s)  = 1  + (−0.357 − 0.619i)3-s − 0.894i·5-s + (1.51 + 0.874i)7-s + (0.244 − 0.422i)9-s + (−0.560 + 0.323i)11-s + (−0.277 − 0.960i)13-s + (−0.554 + 0.319i)15-s + (0.0562 − 0.0974i)17-s + (0.918 + 0.530i)19-s − 1.25i·21-s + (−0.576 − 0.999i)23-s + 0.200·25-s − 1.06·27-s + (−0.278 − 0.482i)29-s − 1.66i·31-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((0.265+0.964i) Λ‾(2−s)\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/2 ΓC(s+1/2) L(s)=((0.265+0.964i) Λ‾(1−s)\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    =    25⋅132^{5} \cdot 13
Sign: 0.265+0.964i0.265 + 0.964i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(257,⋅)\chi_{416} (257, \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 1.09835−0.837218i1.09835 - 0.837218i
L(12)L(\frac12) ≈\approx 1.09835−0.837218i1.09835 - 0.837218i
L(32)L(\frac{3}{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+(1+3.46i)T 1 + (1 + 3.46i)T
good3 1+(0.619+1.07i)T+(−1.5+2.59i)T2 1 + (0.619 + 1.07i)T + (-1.5 + 2.59i)T^{2}
5 1+2iT−5T2 1 + 2iT - 5T^{2}
7 1+(−4.00−2.31i)T+(3.5+6.06i)T2 1 + (-4.00 - 2.31i)T + (3.5 + 6.06i)T^{2}
11 1+(1.85−1.07i)T+(5.5−9.52i)T2 1 + (1.85 - 1.07i)T + (5.5 - 9.52i)T^{2}
17 1+(−0.232+0.401i)T+(−8.5−14.7i)T2 1 + (-0.232 + 0.401i)T + (-8.5 - 14.7i)T^{2}
19 1+(−4.00−2.31i)T+(9.5+16.4i)T2 1 + (-4.00 - 2.31i)T + (9.5 + 16.4i)T^{2}
23 1+(2.76+4.79i)T+(−11.5+19.9i)T2 1 + (2.76 + 4.79i)T + (-11.5 + 19.9i)T^{2}
29 1+(1.5+2.59i)T+(−14.5+25.1i)T2 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2}
31 1+9.25iT−31T2 1 + 9.25iT - 31T^{2}
37 1+(−6.69+3.86i)T+(18.5−32.0i)T2 1 + (-6.69 + 3.86i)T + (18.5 - 32.0i)T^{2}
41 1+(6.23−3.59i)T+(20.5−35.5i)T2 1 + (6.23 - 3.59i)T + (20.5 - 35.5i)T^{2}
43 1+(1.85−3.21i)T+(−21.5−37.2i)T2 1 + (1.85 - 3.21i)T + (-21.5 - 37.2i)T^{2}
47 1−11.7iT−47T2 1 - 11.7iT - 47T^{2}
53 1−2.53T+53T2 1 - 2.53T + 53T^{2}
59 1+(−8.29−4.79i)T+(29.5+51.0i)T2 1 + (-8.29 - 4.79i)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+(4.00−2.31i)T+(33.5−58.0i)T2 1 + (4.00 - 2.31i)T + (33.5 - 58.0i)T^{2}
71 1+(5.57+3.21i)T+(35.5+61.4i)T2 1 + (5.57 + 3.21i)T + (35.5 + 61.4i)T^{2}
73 1−6iT−73T2 1 - 6iT - 73T^{2}
79 1−4.29T+79T2 1 - 4.29T + 79T^{2}
83 1−4.29iT−83T2 1 - 4.29iT - 83T^{2}
89 1+(1.03−0.598i)T+(44.5−77.0i)T2 1 + (1.03 - 0.598i)T + (44.5 - 77.0i)T^{2}
97 1+(−11.8−6.86i)T+(48.5+84.0i)T2 1 + (-11.8 - 6.86i)T + (48.5 + 84.0i)T^{2}
show more
show less
   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

−11.35408886688630642585445331334, −10.08806398786797250892964511995, −9.113259503956614243682656895656, −7.990518267979334447046074562532, −7.69664714416624774970310076032, −6.03428051545010489667513317975, −5.30890435539017129260970331786, −4.38956819989534543368259184009, −2.39144556708733591034295391988, −1.05325020756801103424235098021, 1.79651805371411852427655471626, 3.50126727094041095816937563672, 4.69965266893849253746246853253, 5.33040217324317127803146178408, 6.97660760018063334746861857320, 7.52225638566090454891065034108, 8.603453937559120483965508278068, 9.982815536599453991414719852995, 10.55134966998982832198447014208, 11.27457946594644196168254246240

Graph of the ZZ-function along the critical line