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Properties

Label 2-2548-91.4-c1-0-9
Degree $2$
Conductor $2548$
Sign $-0.936 + 0.350i$
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  + (1.49 + 2.58i)3-s + (0.626 − 0.361i)5-s + (−2.94 + 5.10i)9-s + (−2.23 + 1.29i)11-s + (−0.896 + 3.49i)13-s + (1.86 + 1.07i)15-s − 4.53·17-s + (−6.26 − 3.61i)19-s + 0.199·23-s + (−2.23 + 3.87i)25-s − 8.64·27-s + (2.84 − 4.93i)29-s + (5.36 + 3.09i)31-s + (−6.67 − 3.85i)33-s − 4.64i·37-s + ⋯
L(s)  = 1  + (0.861 + 1.49i)3-s + (0.280 − 0.161i)5-s + (−0.982 + 1.70i)9-s + (−0.674 + 0.389i)11-s + (−0.248 + 0.968i)13-s + (0.482 + 0.278i)15-s − 1.10·17-s + (−1.43 − 0.829i)19-s + 0.0416·23-s + (−0.447 + 0.775i)25-s − 1.66·27-s + (0.529 − 0.916i)29-s + (0.963 + 0.556i)31-s + (−1.16 − 0.670i)33-s − 0.762i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.936 + 0.350i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (1733, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ -0.936 + 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.229709889\)
\(L(\frac12)\) \(\approx\) \(1.229709889\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.896 - 3.49i)T \)
good3 \( 1 + (-1.49 - 2.58i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.626 + 0.361i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.23 - 1.29i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 4.53T + 17T^{2} \)
19 \( 1 + (6.26 + 3.61i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.199T + 23T^{2} \)
29 \( 1 + (-2.84 + 4.93i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.36 - 3.09i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.64iT - 37T^{2} \)
41 \( 1 + (8.26 + 4.77i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.54 - 6.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.838 + 0.484i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.13 + 3.69i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.64iT - 59T^{2} \)
61 \( 1 + (4.36 - 7.56i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.36 + 1.36i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.0784 + 0.0453i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.56 + 2.05i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.96 + 6.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.65iT - 83T^{2} \)
89 \( 1 - 14.1iT - 89T^{2} \)
97 \( 1 + (1.09 - 0.630i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.085633375392236466653034458793, −8.942295795373083270342980786261, −8.092187306293560733463598391963, −7.09262668479667414450108050393, −6.18755332896473968465333505129, −5.01074679023435483699671424907, −4.52398450835297329974871059906, −3.90232629671667715833741839963, −2.66743776707065918551888144917, −2.11885688851900399201851242097, 0.31938338065813431573911505817, 1.70579775142928869257523398432, 2.50464436332491958243725206501, 3.15592258099058824366471298187, 4.41058833677496377278128531029, 5.62316787370461107856123141650, 6.43316365619174643188973176276, 6.87293959087884540957446046356, 7.999155122722020871703334142309, 8.208974077973669116073565232009

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