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Properties

Label 2-832-52.47-c1-0-8
Degree 22
Conductor 832832
Sign 0.916+0.399i0.916 + 0.399i
Analytic cond. 6.643556.64355
Root an. cond. 2.577502.57750
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  − i·3-s + (−1.58 − 1.58i)5-s + (1.58 + 1.58i)7-s + 2·9-s + (2.16 + 2.16i)11-s + (−0.418 + 3.58i)13-s + (−1.58 + 1.58i)15-s + 5.32i·17-s + (5.16 − 5.16i)19-s + (1.58 − 1.58i)21-s + 0.837·23-s − 5i·27-s − 5.16·29-s + (5.16 − 5.16i)31-s + (2.16 − 2.16i)33-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.707 − 0.707i)5-s + (0.597 + 0.597i)7-s + 0.666·9-s + (0.651 + 0.651i)11-s + (−0.116 + 0.993i)13-s + (−0.408 + 0.408i)15-s + 1.29i·17-s + (1.18 − 1.18i)19-s + (0.345 − 0.345i)21-s + 0.174·23-s − 0.962i·27-s − 0.958·29-s + (0.927 − 0.927i)31-s + (0.376 − 0.376i)33-s + ⋯

Functional equation

Λ(s)=(832s/2 ΓC(s) L(s)=((0.916+0.399i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(832s/2 ΓC(s+1/2) L(s)=((0.916+0.399i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 832832    =    26⋅132^{6} \cdot 13
Sign: 0.916+0.399i0.916 + 0.399i
Analytic conductor: 6.643556.64355
Root analytic conductor: 2.577502.57750
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ832(255,⋅)\chi_{832} (255, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 832, ( :1/2), 0.916+0.399i)(2,\ 832,\ (\ :1/2),\ 0.916 + 0.399i)

Particular Values

L(1)L(1) ≈\approx 1.60121−0.333289i1.60121 - 0.333289i
L(12)L(\frac12) ≈\approx 1.60121−0.333289i1.60121 - 0.333289i
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+(0.418−3.58i)T 1 + (0.418 - 3.58i)T
good3 1+iT−3T2 1 + iT - 3T^{2}
5 1+(1.58+1.58i)T+5iT2 1 + (1.58 + 1.58i)T + 5iT^{2}
7 1+(−1.58−1.58i)T+7iT2 1 + (-1.58 - 1.58i)T + 7iT^{2}
11 1+(−2.16−2.16i)T+11iT2 1 + (-2.16 - 2.16i)T + 11iT^{2}
17 1−5.32iT−17T2 1 - 5.32iT - 17T^{2}
19 1+(−5.16+5.16i)T−19iT2 1 + (-5.16 + 5.16i)T - 19iT^{2}
23 1−0.837T+23T2 1 - 0.837T + 23T^{2}
29 1+5.16T+29T2 1 + 5.16T + 29T^{2}
31 1+(−5.16+5.16i)T−31iT2 1 + (-5.16 + 5.16i)T - 31iT^{2}
37 1+(−0.418+0.418i)T−37iT2 1 + (-0.418 + 0.418i)T - 37iT^{2}
41 1+(1.16+1.16i)T+41iT2 1 + (1.16 + 1.16i)T + 41iT^{2}
43 1−5T+43T2 1 - 5T + 43T^{2}
47 1+(−2.74−2.74i)T+47iT2 1 + (-2.74 - 2.74i)T + 47iT^{2}
53 1−9.48T+53T2 1 - 9.48T + 53T^{2}
59 1+(−4−4i)T+59iT2 1 + (-4 - 4i)T + 59iT^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1+(5.32−5.32i)T−67iT2 1 + (5.32 - 5.32i)T - 67iT^{2}
71 1+(−1.58+1.58i)T−71iT2 1 + (-1.58 + 1.58i)T - 71iT^{2}
73 1+(6−6i)T−73iT2 1 + (6 - 6i)T - 73iT^{2}
79 1+15.4iT−79T2 1 + 15.4iT - 79T^{2}
83 1+(12.1−12.1i)T−83iT2 1 + (12.1 - 12.1i)T - 83iT^{2}
89 1+(9.16−9.16i)T−89iT2 1 + (9.16 - 9.16i)T - 89iT^{2}
97 1+(10.1+10.1i)T+97iT2 1 + (10.1 + 10.1i)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

−10.02187270258033471667364694374, −9.149877047018607673086603917155, −8.480253403851347330582991488322, −7.52760048231310368150659583009, −6.93240619363106360047198192706, −5.78345553161091382466960814130, −4.56942151507851981626280540177, −4.06520034774877815047634847791, −2.22953969707436585094543426499, −1.17888513327873646033439619151, 1.10959431488628920150437446405, 3.11570543813214050581105733495, 3.76852924375064055021378055185, 4.79003191676737789047365590490, 5.76902237208018730887263107122, 7.24540711618315172626731583655, 7.42990170117202509222450356877, 8.537027082353804669769299242756, 9.647995358056612608454678818658, 10.27917477054087500789819882562

Graph of the ZZ-function along the critical line