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Properties

Label 2-930-5.4-c1-0-1
Degree 22
Conductor 930930
Sign −0.447+0.894i-0.447 + 0.894i
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 + (−1 + 2i)5-s − 6-s + i·7-s − i·8-s − 9-s + (−2 − i)10-s − 3·11-s − i·12-s + 4i·13-s − 14-s + (−2 − i)15-s + 16-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.447 + 0.894i)5-s − 0.408·6-s + 0.377i·7-s − 0.353i·8-s − 0.333·9-s + (−0.632 − 0.316i)10-s − 0.904·11-s − 0.288i·12-s + 1.10i·13-s − 0.267·14-s + (−0.516 − 0.258i)15-s + 0.250·16-s + ⋯

Functional equation

Λ(s)=(930s/2 ΓC(s) L(s)=((−0.447+0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(930s/2 ΓC(s+1/2) L(s)=((−0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 930930    =    2⋅3⋅5⋅312 \cdot 3 \cdot 5 \cdot 31
Sign: −0.447+0.894i-0.447 + 0.894i
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), −0.447+0.894i)(2,\ 930,\ (\ :1/2),\ -0.447 + 0.894i)

Particular Values

L(1)L(1) ≈\approx 0.305556−0.494401i0.305556 - 0.494401i
L(12)L(\frac12) ≈\approx 0.305556−0.494401i0.305556 - 0.494401i
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+(1−2i)T 1 + (1 - 2i)T
31 1+T 1 + T
good7 1−iT−7T2 1 - iT - 7T^{2}
11 1+3T+11T2 1 + 3T + 11T^{2}
13 1−4iT−13T2 1 - 4iT - 13T^{2}
17 1−17T2 1 - 17T^{2}
19 1−T+19T2 1 - T + 19T^{2}
23 1+5iT−23T2 1 + 5iT - 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
37 1−4iT−37T2 1 - 4iT - 37T^{2}
41 1+10T+41T2 1 + 10T + 41T^{2}
43 1−5iT−43T2 1 - 5iT - 43T^{2}
47 1+8iT−47T2 1 + 8iT - 47T^{2}
53 1+5iT−53T2 1 + 5iT - 53T^{2}
59 1−6T+59T2 1 - 6T + 59T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1−2iT−67T2 1 - 2iT - 67T^{2}
71 1+5T+71T2 1 + 5T + 71T^{2}
73 1−7iT−73T2 1 - 7iT - 73T^{2}
79 1+3T+79T2 1 + 3T + 79T^{2}
83 1−2iT−83T2 1 - 2iT - 83T^{2}
89 1+T+89T2 1 + T + 89T^{2}
97 1+10iT−97T2 1 + 10iT - 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

−10.44008843758896986894137273194, −9.860805618875227593481910657206, −8.800218478491754257952755942119, −8.154227292792793092578435497507, −7.14030689535698297919481576656, −6.48148449414858039901327596220, −5.43887489988290248425580158710, −4.52910691332778202043452752015, −3.54288568469892368614829201748, −2.40531935874045117295226274603, 0.27446055720847916293724345868, 1.51866075989609014854314795593, 2.92225776139589165403907765123, 3.92168080914936600301426973713, 5.12899848982183335966401969341, 5.69711276393135988105786699428, 7.30237574232669532135802547708, 7.86416968838300278318250902741, 8.627607386761118794623801111332, 9.545010160273345635842361833700

Graph of the ZZ-function along the critical line