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Properties

Label 2-416-52.47-c1-0-12
Degree 22
Conductor 416416
Sign −0.957+0.289i-0.957 + 0.289i
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  − 2i·3-s + (−1 − i)5-s + (−1 − i)7-s − 9-s + (−3 − 3i)11-s + (−3 + 2i)13-s + (−2 + 2i)15-s + 4i·17-s + (3 − 3i)19-s + (−2 + 2i)21-s − 3i·25-s − 4i·27-s − 6·29-s + (−3 + 3i)31-s + (−6 + 6i)33-s + ⋯
L(s)  = 1  − 1.15i·3-s + (−0.447 − 0.447i)5-s + (−0.377 − 0.377i)7-s − 0.333·9-s + (−0.904 − 0.904i)11-s + (−0.832 + 0.554i)13-s + (−0.516 + 0.516i)15-s + 0.970i·17-s + (0.688 − 0.688i)19-s + (−0.436 + 0.436i)21-s − 0.600i·25-s − 0.769i·27-s − 1.11·29-s + (−0.538 + 0.538i)31-s + (−1.04 + 1.04i)33-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((−0.957+0.289i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((−0.957+0.289i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: −0.957+0.289i-0.957 + 0.289i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(255,⋅)\chi_{416} (255, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), −0.957+0.289i)(2,\ 416,\ (\ :1/2),\ -0.957 + 0.289i)

Particular Values

L(1)L(1) ≈\approx 0.123168−0.831835i0.123168 - 0.831835i
L(12)L(\frac12) ≈\approx 0.123168−0.831835i0.123168 - 0.831835i
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+(3−2i)T 1 + (3 - 2i)T
good3 1+2iT−3T2 1 + 2iT - 3T^{2}
5 1+(1+i)T+5iT2 1 + (1 + i)T + 5iT^{2}
7 1+(1+i)T+7iT2 1 + (1 + i)T + 7iT^{2}
11 1+(3+3i)T+11iT2 1 + (3 + 3i)T + 11iT^{2}
17 1−4iT−17T2 1 - 4iT - 17T^{2}
19 1+(−3+3i)T−19iT2 1 + (-3 + 3i)T - 19iT^{2}
23 1+23T2 1 + 23T^{2}
29 1+6T+29T2 1 + 6T + 29T^{2}
31 1+(3−3i)T−31iT2 1 + (3 - 3i)T - 31iT^{2}
37 1+(−3+3i)T−37iT2 1 + (-3 + 3i)T - 37iT^{2}
41 1+(−1−i)T+41iT2 1 + (-1 - i)T + 41iT^{2}
43 1−4T+43T2 1 - 4T + 43T^{2}
47 1+(5+5i)T+47iT2 1 + (5 + 5i)T + 47iT^{2}
53 1−6T+53T2 1 - 6T + 53T^{2}
59 1+(7+7i)T+59iT2 1 + (7 + 7i)T + 59iT^{2}
61 1−14T+61T2 1 - 14T + 61T^{2}
67 1+(5−5i)T−67iT2 1 + (5 - 5i)T - 67iT^{2}
71 1+(−5+5i)T−71iT2 1 + (-5 + 5i)T - 71iT^{2}
73 1+(−9+9i)T−73iT2 1 + (-9 + 9i)T - 73iT^{2}
79 1−6iT−79T2 1 - 6iT - 79T^{2}
83 1+(−7+7i)T−83iT2 1 + (-7 + 7i)T - 83iT^{2}
89 1+(−5+5i)T−89iT2 1 + (-5 + 5i)T - 89iT^{2}
97 1+(−13−13i)T+97iT2 1 + (-13 - 13i)T + 97iT^{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

−10.96173750481516300347425184576, −9.905592884860998882103158672412, −8.758970292404432880813068789228, −7.85093021649198076209079208680, −7.21131729277581123489759950423, −6.21889772051936640913666958750, −5.04094630012188367036566922776, −3.67356029911679108549950133231, −2.18834617534372465151181343882, −0.52370447396557550538899867305, 2.61021113341454942397550605695, 3.66367831139067676831793992606, 4.84364698487181914157644063694, 5.58941928845342107136503117268, 7.22634437943545366626965811223, 7.76391438211159487140878230787, 9.365098295939244004374170985556, 9.731584454989350587810646422209, 10.58697352281184241312716992204, 11.43944022927050098461517154926

Graph of the ZZ-function along the critical line