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Properties

Label 2-416-13.9-c1-0-7
Degree 22
Conductor 416416
Sign 0.981−0.189i0.981 - 0.189i
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  + 4.46·5-s + (1.5 + 2.59i)9-s + (−3.23 + 1.59i)13-s + (−2.96 − 5.13i)17-s + 14.9·25-s + (−0.767 + 1.33i)29-s + (−5.69 + 9.86i)37-s + (5.96 − 10.3i)41-s + (6.69 + 11.5i)45-s + (3.5 − 6.06i)49-s − 3.53·53-s + (−7.69 − 13.3i)61-s + (−14.4 + 7.13i)65-s − 10.8·73-s + (−4.5 + 7.79i)81-s + ⋯
L(s)  = 1  + 1.99·5-s + (0.5 + 0.866i)9-s + (−0.896 + 0.443i)13-s + (−0.718 − 1.24i)17-s + 2.98·25-s + (−0.142 + 0.246i)29-s + (−0.936 + 1.62i)37-s + (0.931 − 1.61i)41-s + (0.998 + 1.72i)45-s + (0.5 − 0.866i)49-s − 0.485·53-s + (−0.985 − 1.70i)61-s + (−1.78 + 0.884i)65-s − 1.27·73-s + (−0.5 + 0.866i)81-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((0.981−0.189i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((0.981−0.189i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 0.981−0.189i0.981 - 0.189i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(321,⋅)\chi_{416} (321, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), 0.981−0.189i)(2,\ 416,\ (\ :1/2),\ 0.981 - 0.189i)

Particular Values

L(1)L(1) ≈\approx 1.84949+0.177150i1.84949 + 0.177150i
L(12)L(\frac12) ≈\approx 1.84949+0.177150i1.84949 + 0.177150i
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+(3.23−1.59i)T 1 + (3.23 - 1.59i)T
good3 1+(−1.5−2.59i)T2 1 + (-1.5 - 2.59i)T^{2}
5 1−4.46T+5T2 1 - 4.46T + 5T^{2}
7 1+(−3.5+6.06i)T2 1 + (-3.5 + 6.06i)T^{2}
11 1+(−5.5−9.52i)T2 1 + (-5.5 - 9.52i)T^{2}
17 1+(2.96+5.13i)T+(−8.5+14.7i)T2 1 + (2.96 + 5.13i)T + (-8.5 + 14.7i)T^{2}
19 1+(−9.5+16.4i)T2 1 + (-9.5 + 16.4i)T^{2}
23 1+(−11.5−19.9i)T2 1 + (-11.5 - 19.9i)T^{2}
29 1+(0.767−1.33i)T+(−14.5−25.1i)T2 1 + (0.767 - 1.33i)T + (-14.5 - 25.1i)T^{2}
31 1+31T2 1 + 31T^{2}
37 1+(5.69−9.86i)T+(−18.5−32.0i)T2 1 + (5.69 - 9.86i)T + (-18.5 - 32.0i)T^{2}
41 1+(−5.96+10.3i)T+(−20.5−35.5i)T2 1 + (-5.96 + 10.3i)T + (-20.5 - 35.5i)T^{2}
43 1+(−21.5+37.2i)T2 1 + (-21.5 + 37.2i)T^{2}
47 1+47T2 1 + 47T^{2}
53 1+3.53T+53T2 1 + 3.53T + 53T^{2}
59 1+(−29.5+51.0i)T2 1 + (-29.5 + 51.0i)T^{2}
61 1+(7.69+13.3i)T+(−30.5+52.8i)T2 1 + (7.69 + 13.3i)T + (-30.5 + 52.8i)T^{2}
67 1+(−33.5−58.0i)T2 1 + (-33.5 - 58.0i)T^{2}
71 1+(−35.5+61.4i)T2 1 + (-35.5 + 61.4i)T^{2}
73 1+10.8T+73T2 1 + 10.8T + 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1+(5−8.66i)T+(−44.5−77.0i)T2 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2}
97 1+(9+15.5i)T+(−48.5+84.0i)T2 1 + (9 + 15.5i)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

−11.00349800252223220338889326889, −10.17906569591599664136860016188, −9.560763677699288361840039651690, −8.790906528279219755841431238019, −7.29225986188843574284946001380, −6.56546368161579126123375522011, −5.34646869418283280224301128829, −4.75217347570088163709179678261, −2.64965181453340557889179221113, −1.79290543314879756681242465962, 1.55645804680609255556160859075, 2.73588033518958669744182930141, 4.39491055892099830466380941802, 5.67140035079707015491724778430, 6.26975979927335576912515009282, 7.26904459171443857326305880975, 8.769313739828517293082285312918, 9.479259206124106537865589622193, 10.15226851265646105006239828246, 10.88557293000375298828973814736

Graph of the ZZ-function along the critical line