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Properties

Label 2-416-13.3-c1-0-12
Degree 22
Conductor 416416
Sign −0.477+0.878i-0.477 + 0.878i
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 − 1.73i)3-s + 5-s + (−0.499 + 0.866i)9-s + (−2 − 3.46i)11-s + (2.5 − 2.59i)13-s + (−1 − 1.73i)15-s + (−1.5 + 2.59i)17-s + (1 − 1.73i)19-s + (−1 − 1.73i)23-s − 4·25-s − 4.00·27-s + (−2.5 − 4.33i)29-s + 2·31-s + (−3.99 + 6.92i)33-s + (−2.5 − 4.33i)37-s + ⋯
L(s)  = 1  + (−0.577 − 0.999i)3-s + 0.447·5-s + (−0.166 + 0.288i)9-s + (−0.603 − 1.04i)11-s + (0.693 − 0.720i)13-s + (−0.258 − 0.447i)15-s + (−0.363 + 0.630i)17-s + (0.229 − 0.397i)19-s + (−0.208 − 0.361i)23-s − 0.800·25-s − 0.769·27-s + (−0.464 − 0.804i)29-s + 0.359·31-s + (−0.696 + 1.20i)33-s + (−0.410 − 0.711i)37-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((−0.477+0.878i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((−0.477+0.878i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: −0.477+0.878i-0.477 + 0.878i
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), −0.477+0.878i)(2,\ 416,\ (\ :1/2),\ -0.477 + 0.878i)

Particular Values

L(1)L(1) ≈\approx 0.542525−0.912455i0.542525 - 0.912455i
L(12)L(\frac12) ≈\approx 0.542525−0.912455i0.542525 - 0.912455i
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+(−2.5+2.59i)T 1 + (-2.5 + 2.59i)T
good3 1+(1+1.73i)T+(−1.5+2.59i)T2 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2}
5 1−T+5T2 1 - T + 5T^{2}
7 1+(−3.5−6.06i)T2 1 + (-3.5 - 6.06i)T^{2}
11 1+(2+3.46i)T+(−5.5+9.52i)T2 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2}
17 1+(1.5−2.59i)T+(−8.5−14.7i)T2 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1+1.73i)T+(−9.5−16.4i)T2 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2}
23 1+(1+1.73i)T+(−11.5+19.9i)T2 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2}
29 1+(2.5+4.33i)T+(−14.5+25.1i)T2 1 + (2.5 + 4.33i)T + (-14.5 + 25.1i)T^{2}
31 1−2T+31T2 1 - 2T + 31T^{2}
37 1+(2.5+4.33i)T+(−18.5+32.0i)T2 1 + (2.5 + 4.33i)T + (-18.5 + 32.0i)T^{2}
41 1+(1.5+2.59i)T+(−20.5+35.5i)T2 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2}
43 1+(−2+3.46i)T+(−21.5−37.2i)T2 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2}
47 1−6T+47T2 1 - 6T + 47T^{2}
53 1−13T+53T2 1 - 13T + 53T^{2}
59 1+(6−10.3i)T+(−29.5−51.0i)T2 1 + (6 - 10.3i)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+(−7−12.1i)T+(−33.5+58.0i)T2 1 + (-7 - 12.1i)T + (-33.5 + 58.0i)T^{2}
71 1+(3−5.19i)T+(−35.5−61.4i)T2 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2}
73 1−7T+73T2 1 - 7T + 73T^{2}
79 1−8T+79T2 1 - 8T + 79T^{2}
83 1−4T+83T2 1 - 4T + 83T^{2}
89 1+(7+12.1i)T+(−44.5+77.0i)T2 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2}
97 1+(−1+1.73i)T+(−48.5−84.0i)T2 1 + (-1 + 1.73i)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

−10.98322650556110006452846065851, −10.24188008281258316411059250208, −8.943373161667052862379020486968, −8.057165138715353796603730874785, −7.11200806205607986272802979645, −5.93636381038984198334203286701, −5.67612001872800744029860585719, −3.85092699556815660406124571196, −2.31192757469160141331754803335, −0.74079073947940448674303916649, 2.01982745446997270784132451189, 3.76521666680915787686574047205, 4.78933096749039358130717592127, 5.55272950821372477571782559729, 6.71005873146354384170307873527, 7.80572251307510414354091038004, 9.133719478554666529755097779734, 9.818620437023757232388838948827, 10.49968898232328623053280887750, 11.36303698360904672155054736461

Graph of the ZZ-function along the critical line