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Properties

Label 2-2548-91.9-c1-0-29
Degree $2$
Conductor $2548$
Sign $0.362 + 0.932i$
Analytic cond. $20.3458$
Root an. cond. $4.51064$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.421·3-s + (0.466 − 0.808i)5-s − 2.82·9-s + 0.959·11-s + (3.50 − 0.864i)13-s + (0.196 − 0.340i)15-s + (1.43 − 2.47i)17-s + 2.32·19-s + (−3.51 − 6.08i)23-s + (2.06 + 3.57i)25-s − 2.45·27-s + (−1.84 + 3.18i)29-s + (−0.286 − 0.496i)31-s + 0.404·33-s + (0.734 + 1.27i)37-s + ⋯
L(s)  = 1  + 0.243·3-s + (0.208 − 0.361i)5-s − 0.940·9-s + 0.289·11-s + (0.970 − 0.239i)13-s + (0.0507 − 0.0879i)15-s + (0.346 − 0.600i)17-s + 0.533·19-s + (−0.732 − 1.26i)23-s + (0.412 + 0.715i)25-s − 0.472·27-s + (−0.341 + 0.592i)29-s + (−0.0514 − 0.0891i)31-s + 0.0703·33-s + (0.120 + 0.209i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.362 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.362 + 0.932i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ 0.362 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.825006519\)
\(L(\frac12)\) \(\approx\) \(1.825006519\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
13 \( 1 + (-3.50 + 0.864i)T \)
good3 \( 1 - 0.421T + 3T^{2} \)
5 \( 1 + (-0.466 + 0.808i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 0.959T + 11T^{2} \)
17 \( 1 + (-1.43 + 2.47i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 2.32T + 19T^{2} \)
23 \( 1 + (3.51 + 6.08i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.84 - 3.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.286 + 0.496i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.734 - 1.27i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.42 + 9.39i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.63 + 2.82i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.23 - 2.13i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.81 + 6.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.11 + 5.39i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 7.96T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + (-1.23 - 2.14i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.49 - 6.04i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.93 + 13.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.06T + 83T^{2} \)
89 \( 1 + (8.07 + 13.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.24 + 7.35i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.766938895284323522135440502906, −8.163345210190629533430124000852, −7.29106315974764795660466329138, −6.34387002205453269490338279203, −5.63650960521565248025164527934, −4.92937244810118772264922204621, −3.76938808730666709464803914721, −3.05348677777688990068277012770, −1.93108740442817451232396623356, −0.63314550436035999645068591475, 1.23718970577902190617976156284, 2.41266971783672096046632228079, 3.36457701842827924374720147000, 4.04903215829355983589076611543, 5.28915167537721041187916603987, 6.06545023077443941437559878578, 6.53030355114039450480176757102, 7.80344070582286772518771918085, 8.125050838358834213754152178073, 9.142246132614663045995523349847

Graph of the $Z$-function along the critical line