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Properties

Label 2-2640-5.4-c1-0-57
Degree 22
Conductor 26402640
Sign −0.990+0.139i-0.990 + 0.139i
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 + (0.311 + 2.21i)5-s − 4.90i·7-s − 9-s + 11-s − 4.14i·13-s + (2.21 − 0.311i)15-s − 5.33i·17-s − 5.18·19-s − 4.90·21-s + 4i·23-s + (−4.80 + 1.37i)25-s + i·27-s − 1.80·29-s − 2.62·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.139 + 0.990i)5-s − 1.85i·7-s − 0.333·9-s + 0.301·11-s − 1.15i·13-s + (0.571 − 0.0803i)15-s − 1.29i·17-s − 1.18·19-s − 1.06·21-s + 0.834i·23-s + (−0.961 + 0.275i)25-s + 0.192i·27-s − 0.335·29-s − 0.470·31-s + ⋯

Functional equation

Λ(s)=(2640s/2 ΓC(s) L(s)=((−0.990+0.139i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2640s/2 ΓC(s+1/2) L(s)=((−0.990+0.139i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26402640    =    24⋅3⋅5⋅112^{4} \cdot 3 \cdot 5 \cdot 11
Sign: −0.990+0.139i-0.990 + 0.139i
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), −0.990+0.139i)(2,\ 2640,\ (\ :1/2),\ -0.990 + 0.139i)

Particular Values

L(1)L(1) ≈\approx 0.88139555480.8813955548
L(12)L(\frac12) ≈\approx 0.88139555480.8813955548
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+(−0.311−2.21i)T 1 + (-0.311 - 2.21i)T
11 1−T 1 - T
good7 1+4.90iT−7T2 1 + 4.90iT - 7T^{2}
13 1+4.14iT−13T2 1 + 4.14iT - 13T^{2}
17 1+5.33iT−17T2 1 + 5.33iT - 17T^{2}
19 1+5.18T+19T2 1 + 5.18T + 19T^{2}
23 1−4iT−23T2 1 - 4iT - 23T^{2}
29 1+1.80T+29T2 1 + 1.80T + 29T^{2}
31 1+2.62T+31T2 1 + 2.62T + 31T^{2}
37 1−5.80iT−37T2 1 - 5.80iT - 37T^{2}
41 1−1.80T+41T2 1 - 1.80T + 41T^{2}
43 1−4.90iT−43T2 1 - 4.90iT - 43T^{2}
47 1+7.05iT−47T2 1 + 7.05iT - 47T^{2}
53 1−7.18iT−53T2 1 - 7.18iT - 53T^{2}
59 1−1.67T+59T2 1 - 1.67T + 59T^{2}
61 1−0.755T+61T2 1 - 0.755T + 61T^{2}
67 1−4.85iT−67T2 1 - 4.85iT - 67T^{2}
71 1+0.428T+71T2 1 + 0.428T + 71T^{2}
73 1+12.7iT−73T2 1 + 12.7iT - 73T^{2}
79 1+6.42T+79T2 1 + 6.42T + 79T^{2}
83 1+2.90iT−83T2 1 + 2.90iT - 83T^{2}
89 1+0.622T+89T2 1 + 0.622T + 89T^{2}
97 1+2.75iT−97T2 1 + 2.75iT - 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

−8.190603013709827337523653058503, −7.53628579320893297039126451593, −7.07104601831783766970802977765, −6.45878661838847249585147021706, −5.51618257470267948394476853436, −4.39892290537778238486795716724, −3.52303173437102395124376152919, −2.77514270505209182235749349670, −1.46287185118712345549711804443, −0.27579647668228536578355633447, 1.77532473541198652605608151932, 2.39371914054203298518144493601, 3.86091782906617814168235339385, 4.46594944128980453040777266346, 5.40661109872207471851989109025, 5.96322073996326127071579241637, 6.67463345187058765259299951920, 8.186505105761158373685278597371, 8.600327972004237773635333529809, 9.139802904578349493196110997163

Graph of the ZZ-function along the critical line