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Properties

Label 2-2548-91.51-c1-0-36
Degree $2$
Conductor $2548$
Sign $-0.265 + 0.964i$
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  + (−0.646 − 1.12i)3-s + (2.42 + 1.39i)5-s + (0.663 − 1.14i)9-s + (−0.894 + 0.516i)11-s + (1.79 − 3.12i)13-s − 3.62i·15-s + (−3.47 − 6.02i)17-s + (2.19 + 1.26i)19-s + (1.91 − 3.31i)23-s + (1.41 + 2.45i)25-s − 5.59·27-s − 5.10·29-s + (−3.28 + 1.89i)31-s + (1.15 + 0.668i)33-s + (−5.37 − 3.10i)37-s + ⋯
L(s)  = 1  + (−0.373 − 0.646i)3-s + (1.08 + 0.625i)5-s + (0.221 − 0.382i)9-s + (−0.269 + 0.155i)11-s + (0.498 − 0.866i)13-s − 0.934i·15-s + (−0.843 − 1.46i)17-s + (0.504 + 0.291i)19-s + (0.399 − 0.691i)23-s + (0.283 + 0.490i)25-s − 1.07·27-s − 0.947·29-s + (−0.589 + 0.340i)31-s + (0.201 + 0.116i)33-s + (−0.883 − 0.510i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.265 + 0.964i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ -0.265 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.631345942\)
\(L(\frac12)\) \(\approx\) \(1.631345942\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.79 + 3.12i)T \)
good3 \( 1 + (0.646 + 1.12i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.42 - 1.39i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.894 - 0.516i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.47 + 6.02i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.19 - 1.26i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.91 + 3.31i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.10T + 29T^{2} \)
31 \( 1 + (3.28 - 1.89i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.37 + 3.10i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.990iT - 41T^{2} \)
43 \( 1 - 0.560T + 43T^{2} \)
47 \( 1 + (-6.22 - 3.59i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.589 + 1.02i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.3 + 5.96i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.92 + 11.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.9 - 6.89i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.80iT - 71T^{2} \)
73 \( 1 + (-2.76 + 1.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.66 - 6.35i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.6iT - 83T^{2} \)
89 \( 1 + (-10.3 - 5.94i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.60iT - 97T^{2} \)
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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

−8.817465223685678971461965385490, −7.62945241813893774154822081920, −7.05598536639200086786370392464, −6.41268858195890665448981243316, −5.68566768357019661662208232006, −5.02142859601844271036803068104, −3.66859753443529566543017811438, −2.69515707420621254601574439080, −1.82227211736056268478490712343, −0.55285574257926996945556979252, 1.46681555373175937534649874508, 2.17956309832178880107563434532, 3.72156410900588993217823825890, 4.39591763131123108437421313211, 5.41849699199180869750751303693, 5.68782123378028480119407910179, 6.70251679431864288281456880162, 7.58122720920630288941467461571, 8.717432002620776791420848986587, 9.078285914610475076700582186403

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