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Properties

Label 2-39e2-13.5-c0-0-1
Degree 22
Conductor 15211521
Sign 0.957+0.289i0.957 + 0.289i
Analytic cond. 0.7590770.759077
Root an. cond. 0.8712500.871250
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)2-s − i·4-s + (−1 + i)5-s + 2i·10-s + (1 + i)11-s + 16-s + (1 + i)20-s + 2·22-s − i·25-s + (1 − i)32-s + (−1 + i)41-s − 2i·43-s + (1 − i)44-s + (1 + i)47-s − i·49-s + (−1 − i)50-s + ⋯
L(s)  = 1  + (1 − i)2-s − i·4-s + (−1 + i)5-s + 2i·10-s + (1 + i)11-s + 16-s + (1 + i)20-s + 2·22-s − i·25-s + (1 − i)32-s + (−1 + i)41-s − 2i·43-s + (1 − i)44-s + (1 + i)47-s − i·49-s + (−1 − i)50-s + ⋯

Functional equation

Λ(s)=(1521s/2 ΓC(s) L(s)=((0.957+0.289i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1521s/2 ΓC(s) L(s)=((0.957+0.289i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15211521    =    32⋅1323^{2} \cdot 13^{2}
Sign: 0.957+0.289i0.957 + 0.289i
Analytic conductor: 0.7590770.759077
Root analytic conductor: 0.8712500.871250
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1521(577,⋅)\chi_{1521} (577, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1521, ( :0), 0.957+0.289i)(2,\ 1521,\ (\ :0),\ 0.957 + 0.289i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.6760540351.676054035
L(12)L(\frac12) ≈\approx 1.6760540351.676054035
L(1)L(1) 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)
bad3 1 1
13 1 1
good2 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
5 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
7 1+iT2 1 + iT^{2}
11 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
17 1−T2 1 - T^{2}
19 1−iT2 1 - iT^{2}
23 1−T2 1 - T^{2}
29 1+T2 1 + T^{2}
31 1−iT2 1 - iT^{2}
37 1+iT2 1 + iT^{2}
41 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
43 1+2iT−T2 1 + 2iT - T^{2}
47 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
53 1+T2 1 + T^{2}
59 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
61 1+T2 1 + T^{2}
67 1−iT2 1 - iT^{2}
71 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
73 1+iT2 1 + iT^{2}
79 1+T2 1 + T^{2}
83 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
89 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
97 1−iT2 1 - iT^{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

−10.02635070076401819652027012716, −8.949432082028980492138583646302, −7.84477301945013166274032704027, −7.14977543470036483781347895672, −6.37207147450831222010923368968, −5.14149409566546238989746991570, −4.21864583562343619871079672453, −3.68363712050568623108998423535, −2.81645471924645602486310125732, −1.72380868001794991272043699220, 1.14579219106845792076663418929, 3.27477425483719994674335466637, 4.06493804663884689961184578365, 4.66352069707040162174770065308, 5.59253313514255286898143936021, 6.31019007411589990166802500253, 7.19290765409074862744117176310, 7.984447783007730916841923059948, 8.627780332323383978289809631682, 9.375520392993560886578239447786

Graph of the ZZ-function along the critical line