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Properties

Label 2-2385-265.158-c0-0-2
Degree 22
Conductor 23852385
Sign 0.997−0.0746i0.997 - 0.0746i
Analytic cond. 1.190271.19027
Root an. cond. 1.090991.09099
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  + (−0.437 − 0.437i)2-s − 0.618i·4-s + (0.987 + 0.156i)5-s + (−0.642 + 0.642i)7-s + (−0.707 + 0.707i)8-s + (−0.363 − 0.5i)10-s + (1.39 + 1.39i)13-s + 0.561·14-s + (0.0966 − 0.610i)20-s + (−1.34 + 1.34i)23-s + (0.951 + 0.309i)25-s − 1.22i·26-s + (0.396 + 0.396i)28-s + (0.707 + 0.707i)32-s + (−0.734 + 0.533i)35-s + ⋯
L(s)  = 1  + (−0.437 − 0.437i)2-s − 0.618i·4-s + (0.987 + 0.156i)5-s + (−0.642 + 0.642i)7-s + (−0.707 + 0.707i)8-s + (−0.363 − 0.5i)10-s + (1.39 + 1.39i)13-s + 0.561·14-s + (0.0966 − 0.610i)20-s + (−1.34 + 1.34i)23-s + (0.951 + 0.309i)25-s − 1.22i·26-s + (0.396 + 0.396i)28-s + (0.707 + 0.707i)32-s + (−0.734 + 0.533i)35-s + ⋯

Functional equation

Λ(s)=(2385s/2 ΓC(s) L(s)=((0.997−0.0746i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2385s/2 ΓC(s) L(s)=((0.997−0.0746i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23852385    =    32⋅5⋅533^{2} \cdot 5 \cdot 53
Sign: 0.997−0.0746i0.997 - 0.0746i
Analytic conductor: 1.190271.19027
Root analytic conductor: 1.090991.09099
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2385(2278,⋅)\chi_{2385} (2278, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2385, ( :0), 0.997−0.0746i)(2,\ 2385,\ (\ :0),\ 0.997 - 0.0746i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0284056781.028405678
L(12)L(\frac12) ≈\approx 1.0284056781.028405678
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
5 1+(−0.987−0.156i)T 1 + (-0.987 - 0.156i)T
53 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
good2 1+(0.437+0.437i)T+iT2 1 + (0.437 + 0.437i)T + iT^{2}
7 1+(0.642−0.642i)T−iT2 1 + (0.642 - 0.642i)T - iT^{2}
11 1+T2 1 + T^{2}
13 1+(−1.39−1.39i)T+iT2 1 + (-1.39 - 1.39i)T + iT^{2}
17 1−iT2 1 - iT^{2}
19 1+T2 1 + T^{2}
23 1+(1.34−1.34i)T−iT2 1 + (1.34 - 1.34i)T - iT^{2}
29 1−T2 1 - T^{2}
31 1−T2 1 - T^{2}
37 1+(−1.26+1.26i)T−iT2 1 + (-1.26 + 1.26i)T - iT^{2}
41 1−0.907iT−T2 1 - 0.907iT - T^{2}
43 1+(−0.221−0.221i)T+iT2 1 + (-0.221 - 0.221i)T + iT^{2}
47 1−iT2 1 - iT^{2}
59 1−T2 1 - T^{2}
61 1−T2 1 - T^{2}
67 1+iT2 1 + iT^{2}
71 1+1.97iT−T2 1 + 1.97iT - T^{2}
73 1−iT2 1 - iT^{2}
79 1+T2 1 + T^{2}
83 1+(−0.831+0.831i)T−iT2 1 + (-0.831 + 0.831i)T - iT^{2}
89 1−T2 1 - T^{2}
97 1+(−0.221+0.221i)T−iT2 1 + (-0.221 + 0.221i)T - 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

−9.316015159837988154249358372475, −8.866137120227015698906995226022, −7.75675173442557862601213056299, −6.42637282222043682960932778227, −6.14678153475878358880470179926, −5.54806870930338487024819948148, −4.33297373355927581669530544787, −3.17258460324451259675289896338, −2.11815299901061952138146306401, −1.45566351241097516128094935803, 0.861753963384573585525045963148, 2.49166582088573938116364142689, 3.40661901637917682969035027220, 4.19469278490963464146957102701, 5.48773133708776932264563674822, 6.32299012512808052355137146595, 6.63949916616092686406681582109, 7.81625622099027429746675794044, 8.326182552250648733201068990135, 8.993069395639826313057459752113

Graph of the ZZ-function along the critical line