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Properties

Label 2-2640-5.4-c1-0-17
Degree 22
Conductor 26402640
Sign 0.749−0.662i0.749 - 0.662i
Analytic cond. 21.080521.0805
Root an. cond. 4.591354.59135
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·3-s + (−1.48 − 1.67i)5-s − 1.19i·7-s − 9-s − 11-s − 0.806i·13-s + (1.67 − 1.48i)15-s + 3.76i·17-s − 5.35·19-s + 1.19·21-s + 4i·23-s + (−0.612 + 4.96i)25-s − i·27-s + 4.31·29-s − 0.962·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.662 − 0.749i)5-s − 0.451i·7-s − 0.333·9-s − 0.301·11-s − 0.223i·13-s + (0.432 − 0.382i)15-s + 0.913i·17-s − 1.22·19-s + 0.260·21-s + 0.834i·23-s + (−0.122 + 0.992i)25-s − 0.192i·27-s + 0.800·29-s − 0.172·31-s + ⋯

Functional equation

Λ(s)=(2640s/2 ΓC(s) L(s)=((0.749−0.662i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2640s/2 ΓC(s+1/2) L(s)=((0.749−0.662i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26402640    =    24⋅3⋅5⋅112^{4} \cdot 3 \cdot 5 \cdot 11
Sign: 0.749−0.662i0.749 - 0.662i
Analytic conductor: 21.080521.0805
Root analytic conductor: 4.591354.59135
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2640(529,⋅)\chi_{2640} (529, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2640, ( :1/2), 0.749−0.662i)(2,\ 2640,\ (\ :1/2),\ 0.749 - 0.662i)

Particular Values

L(1)L(1) ≈\approx 1.2344089951.234408995
L(12)L(\frac12) ≈\approx 1.2344089951.234408995
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−iT 1 - iT
5 1+(1.48+1.67i)T 1 + (1.48 + 1.67i)T
11 1+T 1 + T
good7 1+1.19iT−7T2 1 + 1.19iT - 7T^{2}
13 1+0.806iT−13T2 1 + 0.806iT - 13T^{2}
17 1−3.76iT−17T2 1 - 3.76iT - 17T^{2}
19 1+5.35T+19T2 1 + 5.35T + 19T^{2}
23 1−4iT−23T2 1 - 4iT - 23T^{2}
29 1−4.31T+29T2 1 - 4.31T + 29T^{2}
31 1+0.962T+31T2 1 + 0.962T + 31T^{2}
37 1−1.61iT−37T2 1 - 1.61iT - 37T^{2}
41 1−9.08T+41T2 1 - 9.08T + 41T^{2}
43 1−4.41iT−43T2 1 - 4.41iT - 43T^{2}
47 1+12.3iT−47T2 1 + 12.3iT - 47T^{2}
53 1+1.42iT−53T2 1 + 1.42iT - 53T^{2}
59 1−13.2T+59T2 1 - 13.2T + 59T^{2}
61 1+0.0752T+61T2 1 + 0.0752T + 61T^{2}
67 1−2.70iT−67T2 1 - 2.70iT - 67T^{2}
71 1−14.0T+71T2 1 - 14.0T + 71T^{2}
73 1−10.7iT−73T2 1 - 10.7iT - 73T^{2}
79 1−13.9T+79T2 1 - 13.9T + 79T^{2}
83 1+9.89iT−83T2 1 + 9.89iT - 83T^{2}
89 1+16.8T+89T2 1 + 16.8T + 89T^{2}
97 1−11.4iT−97T2 1 - 11.4iT - 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

−8.772725883677591832124471082981, −8.329018101570817948272871827236, −7.61036721483342525558845958042, −6.66890331415702974650305628050, −5.69306686036635602482810992159, −4.94105589999368435064477576306, −4.08364994614778861624053789734, −3.62362379909875747507960830670, −2.26356132652602173684245305026, −0.844845720571783865139902330449, 0.55298553805198929717913163165, 2.27021186737293786338344045180, 2.77368565703586902391053758035, 3.95386488446155801072114187920, 4.78768635491278309561050536293, 5.89284410021160953758037319343, 6.58401394103575596046598262579, 7.21657901981280274939950388017, 7.980851406146128205305365973755, 8.608759176960002325258124045259

Graph of the ZZ-function along the critical line