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Properties

Label 2-416-52.31-c1-0-11
Degree 22
Conductor 416416
Sign −0.289+0.957i-0.289 + 0.957i
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)=((−0.289+0.957i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((−0.289+0.957i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: −0.289+0.957i-0.289 + 0.957i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(31,⋅)\chi_{416} (31, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), −0.289+0.957i)(2,\ 416,\ (\ :1/2),\ -0.289 + 0.957i)

Particular Values

L(1)L(1) ≈\approx 0.742334−1.00037i0.742334 - 1.00037i
L(12)L(\frac12) ≈\approx 0.742334−1.00037i0.742334 - 1.00037i
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

−11.31392185273489285356462307776, −10.12777678751927162915931077554, −8.944229712393368449988817767770, −7.933929834186261852628887955234, −7.17275869981612183839742504189, −6.57592182355326090814732601396, −5.22629001695797034811649077698, −3.82050309870889071469739403267, −2.47536426987382414381381696534, −0.846320689843376080608163233163, 2.00355756099595408409347565738, 4.02296685213288732680880407294, 4.32118410482290499775206239259, 5.49226295140879080486505428328, 6.80045863406213119772349186885, 7.990392797744011416122359405847, 8.933277050891136879932285194976, 9.689401752764033736154683050459, 10.40985395938440876846530180153, 11.50949877339205199146872425380

Graph of the ZZ-function along the critical line