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Properties

Label 2-2548-7.4-c1-0-33
Degree 22
Conductor 25482548
Sign −0.386+0.922i-0.386 + 0.922i
Analytic cond. 20.345820.3458
Root an. cond. 4.510644.51064
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  + (1 − 1.73i)3-s + (−0.5 − 0.866i)5-s + (−0.499 − 0.866i)9-s + (2 − 3.46i)11-s + 13-s − 1.99·15-s + (1 − 1.73i)17-s + (0.5 + 0.866i)19-s + (3.5 + 6.06i)23-s + (2 − 3.46i)25-s + 4.00·27-s − 5·29-s + (4.5 − 7.79i)31-s + (−3.99 − 6.92i)33-s + (1 + 1.73i)37-s + ⋯
L(s)  = 1  + (0.577 − 0.999i)3-s + (−0.223 − 0.387i)5-s + (−0.166 − 0.288i)9-s + (0.603 − 1.04i)11-s + 0.277·13-s − 0.516·15-s + (0.242 − 0.420i)17-s + (0.114 + 0.198i)19-s + (0.729 + 1.26i)23-s + (0.400 − 0.692i)25-s + 0.769·27-s − 0.928·29-s + (0.808 − 1.39i)31-s + (−0.696 − 1.20i)33-s + (0.164 + 0.284i)37-s + ⋯

Functional equation

Λ(s)=(2548s/2 ΓC(s) L(s)=((−0.386+0.922i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2548s/2 ΓC(s+1/2) L(s)=((−0.386+0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25482548    =    22⋅72⋅132^{2} \cdot 7^{2} \cdot 13
Sign: −0.386+0.922i-0.386 + 0.922i
Analytic conductor: 20.345820.3458
Root analytic conductor: 4.510644.51064
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2548(1145,⋅)\chi_{2548} (1145, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2548, ( :1/2), −0.386+0.922i)(2,\ 2548,\ (\ :1/2),\ -0.386 + 0.922i)

Particular Values

L(1)L(1) ≈\approx 2.2402605092.240260509
L(12)L(\frac12) ≈\approx 2.2402605092.240260509
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
7 1 1
13 1−T 1 - T
good3 1+(−1+1.73i)T+(−1.5−2.59i)T2 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2}
5 1+(0.5+0.866i)T+(−2.5+4.33i)T2 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2}
11 1+(−2+3.46i)T+(−5.5−9.52i)T2 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2}
17 1+(−1+1.73i)T+(−8.5−14.7i)T2 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.5−0.866i)T+(−9.5+16.4i)T2 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2}
23 1+(−3.5−6.06i)T+(−11.5+19.9i)T2 1 + (-3.5 - 6.06i)T + (-11.5 + 19.9i)T^{2}
29 1+5T+29T2 1 + 5T + 29T^{2}
31 1+(−4.5+7.79i)T+(−15.5−26.8i)T2 1 + (-4.5 + 7.79i)T + (-15.5 - 26.8i)T^{2}
37 1+(−1−1.73i)T+(−18.5+32.0i)T2 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2}
41 1−2T+41T2 1 - 2T + 41T^{2}
43 1−T+43T2 1 - T + 43T^{2}
47 1+(4.5+7.79i)T+(−23.5+40.7i)T2 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2}
53 1+(1.5−2.59i)T+(−26.5−45.8i)T2 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2}
59 1+(−29.5−51.0i)T2 1 + (-29.5 - 51.0i)T^{2}
61 1+(7+12.1i)T+(−30.5+52.8i)T2 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2}
67 1+(5−8.66i)T+(−33.5−58.0i)T2 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2}
71 1+14T+71T2 1 + 14T + 71T^{2}
73 1+(1.5−2.59i)T+(−36.5−63.2i)T2 1 + (1.5 - 2.59i)T + (-36.5 - 63.2i)T^{2}
79 1+(2.5+4.33i)T+(−39.5+68.4i)T2 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2}
83 1−5T+83T2 1 - 5T + 83T^{2}
89 1+(−4.5−7.79i)T+(−44.5+77.0i)T2 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2}
97 1+T+97T2 1 + T + 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

−8.576557333304831194821038880185, −7.86503275641902387232717258353, −7.33172431260927536375271535051, −6.41395353529576368830825707878, −5.71998758098083371334693730987, −4.67166847417759882508252958057, −3.62155780516508405477539417710, −2.82659703759217071234806557988, −1.65098321519603733686082269949, −0.75395473796557944257600146978, 1.40487108238202274894544768303, 2.78669386246033289589965643685, 3.46543002298796179756053140952, 4.38152068970085890237994398879, 4.85481365142780101930846655449, 6.12892221193525003179226695600, 6.91388535701494826473542892400, 7.59615207306682079744560021213, 8.680865787335957949403779501525, 9.092496202876055546141401646199

Graph of the ZZ-function along the critical line