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Properties

Label 2-930-5.4-c1-0-9
Degree 22
Conductor 930930
Sign −0.270−0.962i-0.270 - 0.962i
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 + (0.605 + 2.15i)5-s + 6-s + 2i·7-s − i·8-s − 9-s + (−2.15 + 0.605i)10-s + 5.21·11-s + i·12-s − 0.789i·13-s − 2·14-s + (2.15 − 0.605i)15-s + 16-s − 0.115i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.270 + 0.962i)5-s + 0.408·6-s + 0.755i·7-s − 0.353i·8-s − 0.333·9-s + (−0.680 + 0.191i)10-s + 1.57·11-s + 0.288i·12-s − 0.219i·13-s − 0.534·14-s + (0.555 − 0.156i)15-s + 0.250·16-s − 0.0279i·17-s + ⋯

Functional equation

Λ(s)=(930s/2 ΓC(s) L(s)=((−0.270−0.962i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(930s/2 ΓC(s+1/2) L(s)=((−0.270−0.962i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 930930    =    2⋅3⋅5⋅312 \cdot 3 \cdot 5 \cdot 31
Sign: −0.270−0.962i-0.270 - 0.962i
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), −0.270−0.962i)(2,\ 930,\ (\ :1/2),\ -0.270 - 0.962i)

Particular Values

L(1)L(1) ≈\approx 0.929058+1.22621i0.929058 + 1.22621i
L(12)L(\frac12) ≈\approx 0.929058+1.22621i0.929058 + 1.22621i
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+(−0.605−2.15i)T 1 + (-0.605 - 2.15i)T
31 1+T 1 + T
good7 1−2iT−7T2 1 - 2iT - 7T^{2}
11 1−5.21T+11T2 1 - 5.21T + 11T^{2}
13 1+0.789iT−13T2 1 + 0.789iT - 13T^{2}
17 1+0.115iT−17T2 1 + 0.115iT - 17T^{2}
19 1+0.115T+19T2 1 + 0.115T + 19T^{2}
23 1−4.42iT−23T2 1 - 4.42iT - 23T^{2}
29 1+4.42T+29T2 1 + 4.42T + 29T^{2}
37 1−6.61iT−37T2 1 - 6.61iT - 37T^{2}
41 1−2T+41T2 1 - 2T + 41T^{2}
43 1−8.61iT−43T2 1 - 8.61iT - 43T^{2}
47 1−0.115iT−47T2 1 - 0.115iT - 47T^{2}
53 1−0.190iT−53T2 1 - 0.190iT - 53T^{2}
59 1−4.19T+59T2 1 - 4.19T + 59T^{2}
61 1−12.4T+61T2 1 - 12.4T + 61T^{2}
67 1−5.82iT−67T2 1 - 5.82iT - 67T^{2}
71 1+13.8T+71T2 1 + 13.8T + 71T^{2}
73 1−10.8iT−73T2 1 - 10.8iT - 73T^{2}
79 1+2.30T+79T2 1 + 2.30T + 79T^{2}
83 1+15.1iT−83T2 1 + 15.1iT - 83T^{2}
89 1−2.23T+89T2 1 - 2.23T + 89T^{2}
97 1+9.21iT−97T2 1 + 9.21iT - 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

−10.08306726759127149652436084652, −9.338121325944569389627327126008, −8.601683032687253795659550243411, −7.56655818534354642520834219571, −6.86196237297376117471208480274, −6.14729178059506032022686174371, −5.49569378576851641124442416441, −4.01845292749549910138937693410, −2.94188179586906768208890435097, −1.57738892623066139293423674829, 0.78776609613215060268633625533, 2.04695849668419524796122432955, 3.77988037861326749413454058666, 4.15923503549313541264775914404, 5.17898751134358155285796112113, 6.22156536663110368332045121950, 7.35461894735942099701605310397, 8.627068987869605110186483465195, 9.078755497866409856467530079804, 9.809777742966365924996867433581

Graph of the ZZ-function along the critical line