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Properties

Label 2-416-13.3-c1-0-11
Degree 22
Conductor 416416
Sign −0.0128+0.999i-0.0128 + 0.999i
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  + (−1.20 − 2.09i)3-s + 2.82·5-s + (2.20 − 3.82i)7-s + (−1.41 + 2.44i)9-s + (1.62 + 2.80i)11-s + (−1 + 3.46i)13-s + (−3.41 − 5.91i)15-s + (2.91 − 5.04i)17-s + (−0.621 + 1.07i)19-s − 10.6·21-s + (0.621 + 1.07i)23-s + 3.00·25-s − 0.414·27-s + (−4.32 − 7.49i)29-s − 5.65·31-s + ⋯
L(s)  = 1  + (−0.696 − 1.20i)3-s + 1.26·5-s + (0.834 − 1.44i)7-s + (−0.471 + 0.816i)9-s + (0.488 + 0.846i)11-s + (−0.277 + 0.960i)13-s + (−0.881 − 1.52i)15-s + (0.706 − 1.22i)17-s + (−0.142 + 0.246i)19-s − 2.32·21-s + (0.129 + 0.224i)23-s + 0.600·25-s − 0.0797·27-s + (−0.803 − 1.39i)29-s − 1.01·31-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((−0.0128+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((−0.0128+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: −0.0128+0.999i-0.0128 + 0.999i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(289,⋅)\chi_{416} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), −0.0128+0.999i)(2,\ 416,\ (\ :1/2),\ -0.0128 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 1.02980−1.04309i1.02980 - 1.04309i
L(12)L(\frac12) ≈\approx 1.02980−1.04309i1.02980 - 1.04309i
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+(1.20+2.09i)T+(−1.5+2.59i)T2 1 + (1.20 + 2.09i)T + (-1.5 + 2.59i)T^{2}
5 1−2.82T+5T2 1 - 2.82T + 5T^{2}
7 1+(−2.20+3.82i)T+(−3.5−6.06i)T2 1 + (-2.20 + 3.82i)T + (-3.5 - 6.06i)T^{2}
11 1+(−1.62−2.80i)T+(−5.5+9.52i)T2 1 + (-1.62 - 2.80i)T + (-5.5 + 9.52i)T^{2}
17 1+(−2.91+5.04i)T+(−8.5−14.7i)T2 1 + (-2.91 + 5.04i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.621−1.07i)T+(−9.5−16.4i)T2 1 + (0.621 - 1.07i)T + (-9.5 - 16.4i)T^{2}
23 1+(−0.621−1.07i)T+(−11.5+19.9i)T2 1 + (-0.621 - 1.07i)T + (-11.5 + 19.9i)T^{2}
29 1+(4.32+7.49i)T+(−14.5+25.1i)T2 1 + (4.32 + 7.49i)T + (-14.5 + 25.1i)T^{2}
31 1+5.65T+31T2 1 + 5.65T + 31T^{2}
37 1+(−3.74−6.48i)T+(−18.5+32.0i)T2 1 + (-3.74 - 6.48i)T + (-18.5 + 32.0i)T^{2}
41 1+(−2.91−5.04i)T+(−20.5+35.5i)T2 1 + (-2.91 - 5.04i)T + (-20.5 + 35.5i)T^{2}
43 1+(−2.03+3.52i)T+(−21.5−37.2i)T2 1 + (-2.03 + 3.52i)T + (-21.5 - 37.2i)T^{2}
47 1+6T+47T2 1 + 6T + 47T^{2}
53 1+2.82T+53T2 1 + 2.82T + 53T^{2}
59 1+(0.621−1.07i)T+(−29.5−51.0i)T2 1 + (0.621 - 1.07i)T + (-29.5 - 51.0i)T^{2}
61 1+(3.5−6.06i)T+(−30.5−52.8i)T2 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2}
67 1+(−6.62−11.4i)T+(−33.5+58.0i)T2 1 + (-6.62 - 11.4i)T + (-33.5 + 58.0i)T^{2}
71 1+(3.62−6.27i)T+(−35.5−61.4i)T2 1 + (3.62 - 6.27i)T + (-35.5 - 61.4i)T^{2}
73 1−12.4T+73T2 1 - 12.4T + 73T^{2}
79 1−6T+79T2 1 - 6T + 79T^{2}
83 1+4T+83T2 1 + 4T + 83T^{2}
89 1+(−1.67−2.89i)T+(−44.5+77.0i)T2 1 + (-1.67 - 2.89i)T + (-44.5 + 77.0i)T^{2}
97 1+(−4.5+7.79i)T+(−48.5−84.0i)T2 1 + (-4.5 + 7.79i)T + (-48.5 - 84.0i)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

−11.26798591332908878027275914545, −9.982773885650309154830260711176, −9.469362538130094024332255790413, −7.76604895823585456810387293220, −7.17435001523400540384268839752, −6.43386664599844950863617194973, −5.35031050011928999273122749769, −4.25630128671750165758581541419, −2.04460082587664742649847813553, −1.18902363818117412411488506714, 1.92858244175304535860187854915, 3.46648177626219165005588939596, 5.03758832816359361039200034102, 5.60551626322680876268635154308, 6.10644366400052375453825882071, 8.020247026217916835361788042620, 9.072847346152842270701154910279, 9.555544843493088140726443421173, 10.81395966862970178321159221668, 10.95296541841046320678756655771

Graph of the ZZ-function along the critical line