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Properties

Label 2-2548-91.9-c1-0-2
Degree 22
Conductor 25482548
Sign −0.586−0.809i-0.586 - 0.809i
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  + 0.652·3-s + (1.14 − 1.98i)5-s − 2.57·9-s − 6.34·11-s + (−3.16 + 1.72i)13-s + (0.747 − 1.29i)15-s + (2.48 − 4.30i)17-s + 4.78·19-s + (1.88 + 3.27i)23-s + (−0.126 − 0.218i)25-s − 3.63·27-s + (−3.86 + 6.69i)29-s + (0.130 + 0.225i)31-s − 4.13·33-s + (2.52 + 4.37i)37-s + ⋯
L(s)  = 1  + 0.376·3-s + (0.512 − 0.887i)5-s − 0.858·9-s − 1.91·11-s + (−0.878 + 0.478i)13-s + (0.192 − 0.334i)15-s + (0.603 − 1.04i)17-s + 1.09·19-s + (0.393 + 0.682i)23-s + (−0.0252 − 0.0436i)25-s − 0.699·27-s + (−0.717 + 1.24i)29-s + (0.0233 + 0.0404i)31-s − 0.719·33-s + (0.415 + 0.720i)37-s + ⋯

Functional equation

Λ(s)=(2548s/2 ΓC(s) L(s)=((−0.586−0.809i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2548s/2 ΓC(s+1/2) L(s)=((−0.586−0.809i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25482548    =    22⋅72⋅132^{2} \cdot 7^{2} \cdot 13
Sign: −0.586−0.809i-0.586 - 0.809i
Analytic conductor: 20.345820.3458
Root analytic conductor: 4.510644.51064
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2548(373,⋅)\chi_{2548} (373, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2548, ( :1/2), −0.586−0.809i)(2,\ 2548,\ (\ :1/2),\ -0.586 - 0.809i)

Particular Values

L(1)L(1) ≈\approx 0.44480606290.4448060629
L(12)L(\frac12) ≈\approx 0.44480606290.4448060629
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+(3.16−1.72i)T 1 + (3.16 - 1.72i)T
good3 1−0.652T+3T2 1 - 0.652T + 3T^{2}
5 1+(−1.14+1.98i)T+(−2.5−4.33i)T2 1 + (-1.14 + 1.98i)T + (-2.5 - 4.33i)T^{2}
11 1+6.34T+11T2 1 + 6.34T + 11T^{2}
17 1+(−2.48+4.30i)T+(−8.5−14.7i)T2 1 + (-2.48 + 4.30i)T + (-8.5 - 14.7i)T^{2}
19 1−4.78T+19T2 1 - 4.78T + 19T^{2}
23 1+(−1.88−3.27i)T+(−11.5+19.9i)T2 1 + (-1.88 - 3.27i)T + (-11.5 + 19.9i)T^{2}
29 1+(3.86−6.69i)T+(−14.5−25.1i)T2 1 + (3.86 - 6.69i)T + (-14.5 - 25.1i)T^{2}
31 1+(−0.130−0.225i)T+(−15.5+26.8i)T2 1 + (-0.130 - 0.225i)T + (-15.5 + 26.8i)T^{2}
37 1+(−2.52−4.37i)T+(−18.5+32.0i)T2 1 + (-2.52 - 4.37i)T + (-18.5 + 32.0i)T^{2}
41 1+(0.101−0.176i)T+(−20.5−35.5i)T2 1 + (0.101 - 0.176i)T + (-20.5 - 35.5i)T^{2}
43 1+(3.84+6.65i)T+(−21.5+37.2i)T2 1 + (3.84 + 6.65i)T + (-21.5 + 37.2i)T^{2}
47 1+(5.05−8.76i)T+(−23.5−40.7i)T2 1 + (5.05 - 8.76i)T + (-23.5 - 40.7i)T^{2}
53 1+(−1.59−2.76i)T+(−26.5+45.8i)T2 1 + (-1.59 - 2.76i)T + (-26.5 + 45.8i)T^{2}
59 1+(7.34−12.7i)T+(−29.5−51.0i)T2 1 + (7.34 - 12.7i)T + (-29.5 - 51.0i)T^{2}
61 1+1.75T+61T2 1 + 1.75T + 61T^{2}
67 1+4.94T+67T2 1 + 4.94T + 67T^{2}
71 1+(−1.93−3.35i)T+(−35.5+61.4i)T2 1 + (-1.93 - 3.35i)T + (-35.5 + 61.4i)T^{2}
73 1+(5.32+9.21i)T+(−36.5+63.2i)T2 1 + (5.32 + 9.21i)T + (-36.5 + 63.2i)T^{2}
79 1+(6.94−12.0i)T+(−39.5−68.4i)T2 1 + (6.94 - 12.0i)T + (-39.5 - 68.4i)T^{2}
83 1+0.589T+83T2 1 + 0.589T + 83T^{2}
89 1+(−1.98−3.43i)T+(−44.5+77.0i)T2 1 + (-1.98 - 3.43i)T + (-44.5 + 77.0i)T^{2}
97 1+(3.13+5.43i)T+(−48.5+84.0i)T2 1 + (3.13 + 5.43i)T + (-48.5 + 84.0i)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

−9.295712994447243143846804016948, −8.436493836054240604289444555698, −7.63214181492084070653746114249, −7.19205345788739433846088035464, −5.65332922451413178572271377114, −5.31685111909294541933395335725, −4.73806776349785007559886682841, −3.13327373024571252527116246187, −2.69860935257819154703465360667, −1.40531879773131658695050530817, 0.12876169783500276359488752841, 2.14362746898206890012162975356, 2.78546741743316626033625531278, 3.39166213523517920380746060590, 4.86732009248702564473367964326, 5.58047764877672797993867365431, 6.17353304338902887168943044711, 7.31370513960660838106458843190, 7.891496418114449198001348209618, 8.402893398555833585042691126457

Graph of the ZZ-function along the critical line