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Properties

Label 2-416-104.69-c1-0-9
Degree 22
Conductor 416416
Sign −0.505+0.862i-0.505 + 0.862i
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  + (−2.36 − 1.36i)3-s + 0.267·5-s + (3 − 1.73i)7-s + (2.23 + 3.86i)9-s + (−1 + 1.73i)11-s + (2.59 − 2.5i)13-s + (−0.633 − 0.366i)15-s + (−3.23 − 5.59i)17-s + (−2.36 − 4.09i)19-s − 9.46·21-s + (−1.09 + 1.90i)23-s − 4.92·25-s − 4.00i·27-s + (−2.59 − 1.5i)29-s − 1.26i·31-s + ⋯
L(s)  = 1  + (−1.36 − 0.788i)3-s + 0.119·5-s + (1.13 − 0.654i)7-s + (0.744 + 1.28i)9-s + (−0.301 + 0.522i)11-s + (0.720 − 0.693i)13-s + (−0.163 − 0.0945i)15-s + (−0.783 − 1.35i)17-s + (−0.542 − 0.940i)19-s − 2.06·21-s + (−0.228 + 0.396i)23-s − 0.985·25-s − 0.769i·27-s + (−0.482 − 0.278i)29-s − 0.227i·31-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((−0.505+0.862i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((−0.505+0.862i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: −0.505+0.862i-0.505 + 0.862i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(17,⋅)\chi_{416} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), −0.505+0.862i)(2,\ 416,\ (\ :1/2),\ -0.505 + 0.862i)

Particular Values

L(1)L(1) ≈\approx 0.403594−0.704249i0.403594 - 0.704249i
L(12)L(\frac12) ≈\approx 0.403594−0.704249i0.403594 - 0.704249i
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+(−2.59+2.5i)T 1 + (-2.59 + 2.5i)T
good3 1+(2.36+1.36i)T+(1.5+2.59i)T2 1 + (2.36 + 1.36i)T + (1.5 + 2.59i)T^{2}
5 1−0.267T+5T2 1 - 0.267T + 5T^{2}
7 1+(−3+1.73i)T+(3.5−6.06i)T2 1 + (-3 + 1.73i)T + (3.5 - 6.06i)T^{2}
11 1+(1−1.73i)T+(−5.5−9.52i)T2 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2}
17 1+(3.23+5.59i)T+(−8.5+14.7i)T2 1 + (3.23 + 5.59i)T + (-8.5 + 14.7i)T^{2}
19 1+(2.36+4.09i)T+(−9.5+16.4i)T2 1 + (2.36 + 4.09i)T + (-9.5 + 16.4i)T^{2}
23 1+(1.09−1.90i)T+(−11.5−19.9i)T2 1 + (1.09 - 1.90i)T + (-11.5 - 19.9i)T^{2}
29 1+(2.59+1.5i)T+(14.5+25.1i)T2 1 + (2.59 + 1.5i)T + (14.5 + 25.1i)T^{2}
31 1+1.26iT−31T2 1 + 1.26iT - 31T^{2}
37 1+(−3.86+6.69i)T+(−18.5−32.0i)T2 1 + (-3.86 + 6.69i)T + (-18.5 - 32.0i)T^{2}
41 1+(1.03+0.598i)T+(20.5+35.5i)T2 1 + (1.03 + 0.598i)T + (20.5 + 35.5i)T^{2}
43 1+(−8.19+4.73i)T+(21.5−37.2i)T2 1 + (-8.19 + 4.73i)T + (21.5 - 37.2i)T^{2}
47 1+3.26iT−47T2 1 + 3.26iT - 47T^{2}
53 1−9.92iT−53T2 1 - 9.92iT - 53T^{2}
59 1+(3.73+6.46i)T+(−29.5+51.0i)T2 1 + (3.73 + 6.46i)T + (-29.5 + 51.0i)T^{2}
61 1+(0.866−0.5i)T+(30.5−52.8i)T2 1 + (0.866 - 0.5i)T + (30.5 - 52.8i)T^{2}
67 1+(5.36−9.29i)T+(−33.5−58.0i)T2 1 + (5.36 - 9.29i)T + (-33.5 - 58.0i)T^{2}
71 1+(−11.0+6.36i)T+(35.5−61.4i)T2 1 + (-11.0 + 6.36i)T + (35.5 - 61.4i)T^{2}
73 1−1.73iT−73T2 1 - 1.73iT - 73T^{2}
79 1+10.3T+79T2 1 + 10.3T + 79T^{2}
83 1−5.46T+83T2 1 - 5.46T + 83T^{2}
89 1+(0.464+0.267i)T+(44.5+77.0i)T2 1 + (0.464 + 0.267i)T + (44.5 + 77.0i)T^{2}
97 1+(−5.19+3i)T+(48.5−84.0i)T2 1 + (-5.19 + 3i)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

−11.17407340771169256239112235856, −10.41642076322190731101810053257, −9.090113357227083482947640547630, −7.69293831761107812313361325677, −7.26415426831664642076149403996, −6.10118608406956304866843771671, −5.21832158275791016213990045980, −4.31347696857637564898181324335, −2.10769982087713304484664145040, −0.63450756198110514803431586676, 1.78258460462793798500784011195, 3.95260671202757555823740954948, 4.76604116671770998357560017450, 5.89520505368743859118808443388, 6.25535885327853927750557500273, 8.031710309829496504434071972468, 8.773774087989420151178556101696, 9.980674843529552834752825905088, 10.89047503317491984597382740425, 11.27085798287056992531937664673

Graph of the ZZ-function along the critical line