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Properties

Label 2-2548-91.30-c1-0-17
Degree $2$
Conductor $2548$
Sign $0.643 + 0.765i$
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  − 0.605·3-s + (−3.48 − 2.01i)5-s − 2.63·9-s + 3.01i·11-s + (1.03 + 3.45i)13-s + (2.11 + 1.21i)15-s + (2.53 − 4.38i)17-s + 3.31i·19-s + (−2.45 − 4.25i)23-s + (5.60 + 9.71i)25-s + 3.41·27-s + (−1.30 + 2.26i)29-s + (−8.86 + 5.11i)31-s − 1.82i·33-s + (−1.76 + 1.02i)37-s + ⋯
L(s)  = 1  − 0.349·3-s + (−1.55 − 0.900i)5-s − 0.877·9-s + 0.909i·11-s + (0.285 + 0.958i)13-s + (0.545 + 0.314i)15-s + (0.614 − 1.06i)17-s + 0.761i·19-s + (−0.511 − 0.886i)23-s + (1.12 + 1.94i)25-s + 0.656·27-s + (−0.243 + 0.421i)29-s + (−1.59 + 0.919i)31-s − 0.318i·33-s + (−0.290 + 0.167i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.643 + 0.765i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (1941, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ 0.643 + 0.765i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6710846765\)
\(L(\frac12)\) \(\approx\) \(0.6710846765\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.03 - 3.45i)T \)
good3 \( 1 + 0.605T + 3T^{2} \)
5 \( 1 + (3.48 + 2.01i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 3.01iT - 11T^{2} \)
17 \( 1 + (-2.53 + 4.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 3.31iT - 19T^{2} \)
23 \( 1 + (2.45 + 4.25i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.30 - 2.26i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.86 - 5.11i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.76 - 1.02i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.252 + 0.145i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.581 + 1.00i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.64 + 2.10i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.74 + 3.02i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.84 - 3.37i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 13.2T + 61T^{2} \)
67 \( 1 + 4.13iT - 67T^{2} \)
71 \( 1 + (1.10 - 0.639i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-12.9 + 7.45i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.45 + 5.99i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.4iT - 83T^{2} \)
89 \( 1 + (0.511 - 0.295i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.94 + 2.85i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.722639610079831158155775050273, −8.082009713110955019091504938372, −7.31951955211937855641090244151, −6.65949500311770359550624017973, −5.40224146704504512395703896402, −4.88652456031013094910494971603, −4.02039036807863898897089941785, −3.29791698587464063649556136938, −1.80533337343142112157151878977, −0.41104898762986041707962024922, 0.64276366222519664525810203679, 2.55889817464157630651148417071, 3.59568360903184840803012659573, 3.75096454430883935203744346952, 5.27078029917858992325624176618, 5.88985137049404324602837030232, 6.71029666761524988296470659311, 7.67841815089566583086028485781, 8.084326513952875366014526399660, 8.723074717159982645813843794924

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