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Properties

Label 2-1530-5.4-c1-0-17
Degree 22
Conductor 15301530
Sign −0.662−0.749i-0.662 - 0.749i
Analytic cond. 12.217112.2171
Root an. cond. 3.495293.49529
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 − 4-s + (1.48 + 1.67i)5-s + 3.35i·7-s − i·8-s + (−1.67 + 1.48i)10-s + 4·11-s − 0.387i·13-s − 3.35·14-s + 16-s + i·17-s + 5.92·19-s + (−1.48 − 1.67i)20-s + 4i·22-s + 1.35i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.662 + 0.749i)5-s + 1.26i·7-s − 0.353i·8-s + (−0.529 + 0.468i)10-s + 1.20·11-s − 0.107i·13-s − 0.895·14-s + 0.250·16-s + 0.242i·17-s + 1.35·19-s + (−0.331 − 0.374i)20-s + 0.852i·22-s + 0.281i·23-s + ⋯

Functional equation

Λ(s)=(1530s/2 ΓC(s) L(s)=((−0.662−0.749i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1530s/2 ΓC(s+1/2) L(s)=((−0.662−0.749i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15301530    =    2⋅32⋅5⋅172 \cdot 3^{2} \cdot 5 \cdot 17
Sign: −0.662−0.749i-0.662 - 0.749i
Analytic conductor: 12.217112.2171
Root analytic conductor: 3.495293.49529
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1530(919,⋅)\chi_{1530} (919, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1530, ( :1/2), −0.662−0.749i)(2,\ 1530,\ (\ :1/2),\ -0.662 - 0.749i)

Particular Values

L(1)L(1) ≈\approx 1.9377428211.937742821
L(12)L(\frac12) ≈\approx 1.9377428211.937742821
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 1
5 1+(−1.48−1.67i)T 1 + (-1.48 - 1.67i)T
17 1−iT 1 - iT
good7 1−3.35iT−7T2 1 - 3.35iT - 7T^{2}
11 1−4T+11T2 1 - 4T + 11T^{2}
13 1+0.387iT−13T2 1 + 0.387iT - 13T^{2}
19 1−5.92T+19T2 1 - 5.92T + 19T^{2}
23 1−1.35iT−23T2 1 - 1.35iT - 23T^{2}
29 1+2.96T+29T2 1 + 2.96T + 29T^{2}
31 1−5.35T+31T2 1 - 5.35T + 31T^{2}
37 1+1.03iT−37T2 1 + 1.03iT - 37T^{2}
41 1+6.31T+41T2 1 + 6.31T + 41T^{2}
43 1+4.38iT−43T2 1 + 4.38iT - 43T^{2}
47 1−6.70iT−47T2 1 - 6.70iT - 47T^{2}
53 1+11.1iT−53T2 1 + 11.1iT - 53T^{2}
59 1+9.53T+59T2 1 + 9.53T + 59T^{2}
61 1−14.0T+61T2 1 - 14.0T + 61T^{2}
67 1+7.61iT−67T2 1 + 7.61iT - 67T^{2}
71 1+13.2T+71T2 1 + 13.2T + 71T^{2}
73 1−11.6iT−73T2 1 - 11.6iT - 73T^{2}
79 1+14.4T+79T2 1 + 14.4T + 79T^{2}
83 1−5.92iT−83T2 1 - 5.92iT - 83T^{2}
89 1−10.7T+89T2 1 - 10.7T + 89T^{2}
97 1−17.5iT−97T2 1 - 17.5iT - 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

−9.527399603757999687257326440654, −9.040347504292444863460320207989, −8.165400000184640037247434709725, −7.15877991994986797995821331516, −6.46997230474933666714812495463, −5.76231668060962214573138593921, −5.13729695385954344441267656852, −3.76422359545536209378676651322, −2.81513284846855175162324334167, −1.56881760973680539037186930177, 0.860597607670620124714320757569, 1.60463336513277830042178161428, 3.08776856048981778753720868911, 4.09889530082656828826569121463, 4.73702336966055990540779957230, 5.77482879594730413168279381917, 6.76114292991961032536968238200, 7.60548849602754104534632598415, 8.641473318806026633229498740686, 9.330110916768530596754491148649

Graph of the ZZ-function along the critical line