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Properties

Label 2-2548-364.111-c0-0-0
Degree $2$
Conductor $2548$
Sign $-0.998 + 0.0545i$
Analytic cond. $1.27161$
Root an. cond. $1.12766$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−1.40 + 1.40i)5-s + (−0.707 + 0.707i)8-s + (0.5 + 0.866i)9-s + (0.991 − 1.71i)10-s + (−0.793 + 0.608i)13-s + (0.500 − 0.866i)16-s + (0.793 + 1.37i)17-s + (−0.707 − 0.707i)18-s + (−0.513 + 1.91i)20-s − 2.93i·25-s + (0.608 − 0.793i)26-s + (−0.258 + 0.448i)29-s + (−0.258 + 0.965i)32-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−1.40 + 1.40i)5-s + (−0.707 + 0.707i)8-s + (0.5 + 0.866i)9-s + (0.991 − 1.71i)10-s + (−0.793 + 0.608i)13-s + (0.500 − 0.866i)16-s + (0.793 + 1.37i)17-s + (−0.707 − 0.707i)18-s + (−0.513 + 1.91i)20-s − 2.93i·25-s + (0.608 − 0.793i)26-s + (−0.258 + 0.448i)29-s + (−0.258 + 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.998 + 0.0545i$
Analytic conductor: \(1.27161\)
Root analytic conductor: \(1.12766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :0),\ -0.998 + 0.0545i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3904319879\)
\(L(\frac12)\) \(\approx\) \(0.3904319879\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 \)
13 \( 1 + (0.793 - 0.608i)T \)
good3 \( 1 + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (1.40 - 1.40i)T - iT^{2} \)
11 \( 1 + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (-0.793 - 1.37i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + (0.315 + 1.17i)T + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + 1.73T + T^{2} \)
59 \( 1 + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.226 + 0.130i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.866 + 0.5i)T^{2} \)
73 \( 1 + (-0.184 - 0.184i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.739 - 0.198i)T + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (1.78 + 0.478i)T + (0.866 + 0.5i)T^{2} \)
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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

−9.592969410540477800283248903744, −8.298179554038446848287565320903, −8.022832856392555260441679290846, −7.19589186604808491353001541428, −6.87388944915650987973467079644, −5.91030464633965317208448688707, −4.70182389714742952532371361973, −3.71028747131820337212416522086, −2.79675199558031077806056402896, −1.76135233366409321697482164667, 0.38389976455030492376451359296, 1.29165405745366336224984248903, 2.92235562657623811810473161668, 3.74363959757895551042122443108, 4.62365024439606630401325610951, 5.49225271149743041363965044976, 6.75387166826711508472607203113, 7.64810021241111156176629287564, 7.81140638433921627663610584326, 8.749945005782087993352422478244

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