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Properties

Label 2-275-11.3-c1-0-3
Degree 22
Conductor 275275
Sign −0.941−0.335i-0.941 - 0.335i
Analytic cond. 2.195882.19588
Root an. cond. 1.481851.48185
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  + (−0.596 + 0.433i)2-s + (0.868 + 2.67i)3-s + (−0.449 + 1.38i)4-s + (−1.67 − 1.21i)6-s + (−0.318 + 0.980i)7-s + (−0.787 − 2.42i)8-s + (−3.96 + 2.88i)9-s + (1.93 + 2.69i)11-s − 4.09·12-s + (2.79 − 2.02i)13-s + (−0.235 − 0.723i)14-s + (−0.834 − 0.606i)16-s + (1.94 + 1.40i)17-s + (1.11 − 3.44i)18-s + (−2.36 − 7.29i)19-s + ⋯
L(s)  = 1  + (−0.421 + 0.306i)2-s + (0.501 + 1.54i)3-s + (−0.224 + 0.692i)4-s + (−0.684 − 0.497i)6-s + (−0.120 + 0.370i)7-s + (−0.278 − 0.857i)8-s + (−1.32 + 0.961i)9-s + (0.583 + 0.811i)11-s − 1.18·12-s + (0.773 − 0.562i)13-s + (−0.0628 − 0.193i)14-s + (−0.208 − 0.151i)16-s + (0.470 + 0.341i)17-s + (0.263 − 0.811i)18-s + (−0.543 − 1.67i)19-s + ⋯

Functional equation

Λ(s)=(275s/2 ΓC(s) L(s)=((−0.941−0.335i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(275s/2 ΓC(s+1/2) L(s)=((−0.941−0.335i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 275275    =    52⋅115^{2} \cdot 11
Sign: −0.941−0.335i-0.941 - 0.335i
Analytic conductor: 2.195882.19588
Root analytic conductor: 1.481851.48185
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ275(201,⋅)\chi_{275} (201, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 275, ( :1/2), −0.941−0.335i)(2,\ 275,\ (\ :1/2),\ -0.941 - 0.335i)

Particular Values

L(1)L(1) ≈\approx 0.184054+1.06372i0.184054 + 1.06372i
L(12)L(\frac12) ≈\approx 0.184054+1.06372i0.184054 + 1.06372i
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)
bad5 1 1
11 1+(−1.93−2.69i)T 1 + (-1.93 - 2.69i)T
good2 1+(0.596−0.433i)T+(0.618−1.90i)T2 1 + (0.596 - 0.433i)T + (0.618 - 1.90i)T^{2}
3 1+(−0.868−2.67i)T+(−2.42+1.76i)T2 1 + (-0.868 - 2.67i)T + (-2.42 + 1.76i)T^{2}
7 1+(0.318−0.980i)T+(−5.66−4.11i)T2 1 + (0.318 - 0.980i)T + (-5.66 - 4.11i)T^{2}
13 1+(−2.79+2.02i)T+(4.01−12.3i)T2 1 + (-2.79 + 2.02i)T + (4.01 - 12.3i)T^{2}
17 1+(−1.94−1.40i)T+(5.25+16.1i)T2 1 + (-1.94 - 1.40i)T + (5.25 + 16.1i)T^{2}
19 1+(2.36+7.29i)T+(−15.3+11.1i)T2 1 + (2.36 + 7.29i)T + (-15.3 + 11.1i)T^{2}
23 1+2.45T+23T2 1 + 2.45T + 23T^{2}
29 1+(1.83−5.66i)T+(−23.4−17.0i)T2 1 + (1.83 - 5.66i)T + (-23.4 - 17.0i)T^{2}
31 1+(−2.98+2.16i)T+(9.57−29.4i)T2 1 + (-2.98 + 2.16i)T + (9.57 - 29.4i)T^{2}
37 1+(1.84−5.66i)T+(−29.9−21.7i)T2 1 + (1.84 - 5.66i)T + (-29.9 - 21.7i)T^{2}
41 1+(−1.21−3.74i)T+(−33.1+24.0i)T2 1 + (-1.21 - 3.74i)T + (-33.1 + 24.0i)T^{2}
43 1−7.64T+43T2 1 - 7.64T + 43T^{2}
47 1+(1.80+5.55i)T+(−38.0+27.6i)T2 1 + (1.80 + 5.55i)T + (-38.0 + 27.6i)T^{2}
53 1+(9.58−6.96i)T+(16.3−50.4i)T2 1 + (9.58 - 6.96i)T + (16.3 - 50.4i)T^{2}
59 1+(−0.910+2.80i)T+(−47.7−34.6i)T2 1 + (-0.910 + 2.80i)T + (-47.7 - 34.6i)T^{2}
61 1+(2.00+1.45i)T+(18.8+58.0i)T2 1 + (2.00 + 1.45i)T + (18.8 + 58.0i)T^{2}
67 1−6.14T+67T2 1 - 6.14T + 67T^{2}
71 1+(1.63+1.18i)T+(21.9+67.5i)T2 1 + (1.63 + 1.18i)T + (21.9 + 67.5i)T^{2}
73 1+(−0.255+0.785i)T+(−59.0−42.9i)T2 1 + (-0.255 + 0.785i)T + (-59.0 - 42.9i)T^{2}
79 1+(−9.77+7.09i)T+(24.4−75.1i)T2 1 + (-9.77 + 7.09i)T + (24.4 - 75.1i)T^{2}
83 1+(−1.30−0.946i)T+(25.6+78.9i)T2 1 + (-1.30 - 0.946i)T + (25.6 + 78.9i)T^{2}
89 1−8.16T+89T2 1 - 8.16T + 89T^{2}
97 1+(−1.97+1.43i)T+(29.9−92.2i)T2 1 + (-1.97 + 1.43i)T + (29.9 - 92.2i)T^{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

−12.29256209987742689583671515435, −11.11483130954298291332237000572, −10.14792791172675280165156760288, −9.246491536242774460199371178760, −8.802126412713698312571194093374, −7.78616205265615217417608876208, −6.43508092021702760568146445487, −4.87341144665704538857633196797, −3.95182034230035503009936128645, −2.94130066996149004232867042177, 0.975830049211209716124609238278, 2.07654363690590482183104590860, 3.79222470031274135295405839594, 5.85716522758775307815197073030, 6.44018370702073265930237664388, 7.77033148578557368954739801480, 8.504240718663721882443289200845, 9.410162570097340852928621992647, 10.56273915047325488227954808302, 11.55608969893853194674985720476

Graph of the ZZ-function along the critical line