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Properties

Label 2-930-5.4-c1-0-14
Degree 22
Conductor 930930
Sign 0.999−0.0235i0.999 - 0.0235i
Analytic cond. 7.426087.42608
Root an. cond. 2.725082.72508
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·2-s − i·3-s − 4-s + (−2.23 + 0.0526i)5-s + 6-s + 2i·7-s − i·8-s − 9-s + (−0.0526 − 2.23i)10-s − 0.470·11-s + i·12-s − 6.47i·13-s − 2·14-s + (0.0526 + 2.23i)15-s + 16-s + 7.04i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.999 + 0.0235i)5-s + 0.408·6-s + 0.755i·7-s − 0.353i·8-s − 0.333·9-s + (−0.0166 − 0.706i)10-s − 0.141·11-s + 0.288i·12-s − 1.79i·13-s − 0.534·14-s + (0.0135 + 0.577i)15-s + 0.250·16-s + 1.70i·17-s + ⋯

Functional equation

Λ(s)=(930s/2 ΓC(s) L(s)=((0.999−0.0235i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0235i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(930s/2 ΓC(s+1/2) L(s)=((0.999−0.0235i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0235i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 930930    =    2⋅3⋅5⋅312 \cdot 3 \cdot 5 \cdot 31
Sign: 0.999−0.0235i0.999 - 0.0235i
Analytic conductor: 7.426087.42608
Root analytic conductor: 2.725082.72508
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ930(559,⋅)\chi_{930} (559, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 930, ( :1/2), 0.999−0.0235i)(2,\ 930,\ (\ :1/2),\ 0.999 - 0.0235i)

Particular Values

L(1)L(1) ≈\approx 1.15891+0.0136488i1.15891 + 0.0136488i
L(12)L(\frac12) ≈\approx 1.15891+0.0136488i1.15891 + 0.0136488i
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−iT 1 - iT
3 1+iT 1 + iT
5 1+(2.23−0.0526i)T 1 + (2.23 - 0.0526i)T
31 1+T 1 + T
good7 1−2iT−7T2 1 - 2iT - 7T^{2}
11 1+0.470T+11T2 1 + 0.470T + 11T^{2}
13 1+6.47iT−13T2 1 + 6.47iT - 13T^{2}
17 1−7.04iT−17T2 1 - 7.04iT - 17T^{2}
19 1−7.04T+19T2 1 - 7.04T + 19T^{2}
23 1+6.94iT−23T2 1 + 6.94iT - 23T^{2}
29 1−6.94T+29T2 1 - 6.94T + 29T^{2}
37 1+1.78iT−37T2 1 + 1.78iT - 37T^{2}
41 1−2T+41T2 1 - 2T + 41T^{2}
43 1−0.210iT−43T2 1 - 0.210iT - 43T^{2}
47 1+7.04iT−47T2 1 + 7.04iT - 47T^{2}
53 1−3.15iT−53T2 1 - 3.15iT - 53T^{2}
59 1−7.15T+59T2 1 - 7.15T + 59T^{2}
61 1−11.2T+61T2 1 - 11.2T + 61T^{2}
67 1+8.26iT−67T2 1 + 8.26iT - 67T^{2}
71 1−0.260T+71T2 1 - 0.260T + 71T^{2}
73 1+11.8iT−73T2 1 + 11.8iT - 73T^{2}
79 1−1.89T+79T2 1 - 1.89T + 79T^{2}
83 1−11.7iT−83T2 1 - 11.7iT - 83T^{2}
89 1+12.0T+89T2 1 + 12.0T + 89T^{2}
97 1+3.52iT−97T2 1 + 3.52iT - 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

−10.12461864792402564012301247190, −8.749850019418616354169742231637, −8.230007075327193991581821151282, −7.70603399963008599449560404552, −6.71097129984658726387873078418, −5.79443270245837435101869813423, −5.05774721293566686167443644511, −3.72840688322061401172338255051, −2.73154872617466493573894720637, −0.75028415678710255794022844439, 1.01461484730051513890731435765, 2.82763970041976481777680247682, 3.76122966367400274574582673828, 4.50416628739632516664040818881, 5.28579331807154264471005186648, 6.97300332411234527161313936242, 7.46318202480501323994333836035, 8.616980038011703230432543684148, 9.489031096826429258656131482698, 9.932713248122373518705156459661

Graph of the ZZ-function along the critical line