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Properties

Label 2-924-21.20-c1-0-23
Degree 22
Conductor 924924
Sign 0.654+0.755i0.654 + 0.755i
Analytic cond. 7.378177.37817
Root an. cond. 2.716282.71628
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  + 1.73·3-s − 1.73·5-s + (2 − 1.73i)7-s + 2.99·9-s − i·11-s − 5.19i·13-s − 2.99·15-s + 1.73i·19-s + (3.46 − 2.99i)21-s − 6i·23-s − 2.00·25-s + 5.19·27-s + 9i·29-s − 3.46i·31-s − 1.73i·33-s + ⋯
L(s)  = 1  + 1.00·3-s − 0.774·5-s + (0.755 − 0.654i)7-s + 0.999·9-s − 0.301i·11-s − 1.44i·13-s − 0.774·15-s + 0.397i·19-s + (0.755 − 0.654i)21-s − 1.25i·23-s − 0.400·25-s + 1.00·27-s + 1.67i·29-s − 0.622i·31-s − 0.301i·33-s + ⋯

Functional equation

Λ(s)=(924s/2 ΓC(s) L(s)=((0.654+0.755i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 924 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(924s/2 ΓC(s+1/2) L(s)=((0.654+0.755i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 924 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 924924    =    22⋅3⋅7⋅112^{2} \cdot 3 \cdot 7 \cdot 11
Sign: 0.654+0.755i0.654 + 0.755i
Analytic conductor: 7.378177.37817
Root analytic conductor: 2.716282.71628
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ924(881,⋅)\chi_{924} (881, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 924, ( :1/2), 0.654+0.755i)(2,\ 924,\ (\ :1/2),\ 0.654 + 0.755i)

Particular Values

L(1)L(1) ≈\approx 1.91481−0.874785i1.91481 - 0.874785i
L(12)L(\frac12) ≈\approx 1.91481−0.874785i1.91481 - 0.874785i
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
3 1−1.73T 1 - 1.73T
7 1+(−2+1.73i)T 1 + (-2 + 1.73i)T
11 1+iT 1 + iT
good5 1+1.73T+5T2 1 + 1.73T + 5T^{2}
13 1+5.19iT−13T2 1 + 5.19iT - 13T^{2}
17 1+17T2 1 + 17T^{2}
19 1−1.73iT−19T2 1 - 1.73iT - 19T^{2}
23 1+6iT−23T2 1 + 6iT - 23T^{2}
29 1−9iT−29T2 1 - 9iT - 29T^{2}
31 1+3.46iT−31T2 1 + 3.46iT - 31T^{2}
37 1−7T+37T2 1 - 7T + 37T^{2}
41 1+6.92T+41T2 1 + 6.92T + 41T^{2}
43 1−10T+43T2 1 - 10T + 43T^{2}
47 1−8.66T+47T2 1 - 8.66T + 47T^{2}
53 1−53T2 1 - 53T^{2}
59 1−1.73T+59T2 1 - 1.73T + 59T^{2}
61 1+13.8iT−61T2 1 + 13.8iT - 61T^{2}
67 1−5T+67T2 1 - 5T + 67T^{2}
71 1−12iT−71T2 1 - 12iT - 71T^{2}
73 1−8.66iT−73T2 1 - 8.66iT - 73T^{2}
79 1+10T+79T2 1 + 10T + 79T^{2}
83 1+13.8T+83T2 1 + 13.8T + 83T^{2}
89 1+6.92T+89T2 1 + 6.92T + 89T^{2}
97 1−17.3iT−97T2 1 - 17.3iT - 97T^{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

−10.07656944216168449215548605324, −8.884902018608363511758220283583, −8.118239792575951093321210441714, −7.75149155864955378651775263387, −6.86436422398206041886817471612, −5.46109175287522224955503737677, −4.34700557052699451998718203702, −3.63004634186886233667922654067, −2.55775686188934470944008093718, −0.976419698040574150863647470160, 1.66708033361650054981379488567, 2.66142579194151408955497801770, 4.02402424101348366872639697135, 4.51110622078658723638201313613, 5.87050646086395433828070532515, 7.16790430783297918888731522718, 7.68240907521243273707105426191, 8.559162039066090908781738931163, 9.199037023102744143216440081137, 9.931888685906614325762686253754

Graph of the ZZ-function along the critical line