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Properties

Label 2-2385-265.52-c0-0-4
Degree 22
Conductor 23852385
Sign 0.379−0.925i0.379 - 0.925i
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 + (−0.156 + 0.987i)5-s + (1.26 + 1.26i)7-s + (0.707 + 0.707i)8-s + (0.363 + 0.5i)10-s + (0.221 − 0.221i)13-s + 1.10·14-s + (−0.610 − 0.0966i)20-s + (−1.34 − 1.34i)23-s + (−0.951 − 0.309i)25-s − 0.193i·26-s + (−0.778 + 0.778i)28-s + (−0.707 + 0.707i)32-s + (−1.44 + 1.04i)35-s + ⋯
L(s)  = 1  + (0.437 − 0.437i)2-s + 0.618i·4-s + (−0.156 + 0.987i)5-s + (1.26 + 1.26i)7-s + (0.707 + 0.707i)8-s + (0.363 + 0.5i)10-s + (0.221 − 0.221i)13-s + 1.10·14-s + (−0.610 − 0.0966i)20-s + (−1.34 − 1.34i)23-s + (−0.951 − 0.309i)25-s − 0.193i·26-s + (−0.778 + 0.778i)28-s + (−0.707 + 0.707i)32-s + (−1.44 + 1.04i)35-s + ⋯

Functional equation

Λ(s)=(2385s/2 ΓC(s) L(s)=((0.379−0.925i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2385s/2 ΓC(s) L(s)=((0.379−0.925i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23852385    =    32⋅5⋅533^{2} \cdot 5 \cdot 53
Sign: 0.379−0.925i0.379 - 0.925i
Analytic conductor: 1.190271.19027
Root analytic conductor: 1.090991.09099
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2385(847,⋅)\chi_{2385} (847, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2385, ( :0), 0.379−0.925i)(2,\ 2385,\ (\ :0),\ 0.379 - 0.925i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.6877577191.687757719
L(12)L(\frac12) ≈\approx 1.6877577191.687757719
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+(0.156−0.987i)T 1 + (0.156 - 0.987i)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+(−1.26−1.26i)T+iT2 1 + (-1.26 - 1.26i)T + iT^{2}
11 1+T2 1 + T^{2}
13 1+(−0.221+0.221i)T−iT2 1 + (-0.221 + 0.221i)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+(0.642+0.642i)T+iT2 1 + (0.642 + 0.642i)T + iT^{2}
41 1+1.78iT−T2 1 + 1.78iT - T^{2}
43 1+(−1.39+1.39i)T−iT2 1 + (-1.39 + 1.39i)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+0.312iT−T2 1 + 0.312iT - 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+(−1.39−1.39i)T+iT2 1 + (-1.39 - 1.39i)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

−9.055273796735032545594768499325, −8.481529284171430259071170163126, −7.82500924155946355986077508406, −7.14471598201569218059913294655, −6.01551838654121387425615683442, −5.33647627612133949719810826209, −4.32517546098766729924469453015, −3.61169528806742828997469086691, −2.36797251755151925912483369283, −2.17730721520262384172828419723, 1.10274054816285054716898597743, 1.74865919950082576267216599250, 3.76463554041042776346932577052, 4.40242660934973347767687651986, 4.96108457612124592451338789844, 5.74486711442119976075857777817, 6.61654900199044980084742111939, 7.73575353949359439774309134815, 7.84622385284706663550499115894, 8.973052293706699991244518575880

Graph of the ZZ-function along the critical line