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Properties

Label 2-2548-91.9-c1-0-2
Degree $2$
Conductor $2548$
Sign $-0.586 - 0.809i$
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  + 0.652·3-s + (1.14 − 1.98i)5-s − 2.57·9-s − 6.34·11-s + (−3.16 + 1.72i)13-s + (0.747 − 1.29i)15-s + (2.48 − 4.30i)17-s + 4.78·19-s + (1.88 + 3.27i)23-s + (−0.126 − 0.218i)25-s − 3.63·27-s + (−3.86 + 6.69i)29-s + (0.130 + 0.225i)31-s − 4.13·33-s + (2.52 + 4.37i)37-s + ⋯
L(s)  = 1  + 0.376·3-s + (0.512 − 0.887i)5-s − 0.858·9-s − 1.91·11-s + (−0.878 + 0.478i)13-s + (0.192 − 0.334i)15-s + (0.603 − 1.04i)17-s + 1.09·19-s + (0.393 + 0.682i)23-s + (−0.0252 − 0.0436i)25-s − 0.699·27-s + (−0.717 + 1.24i)29-s + (0.0233 + 0.0404i)31-s − 0.719·33-s + (0.415 + 0.720i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.586 - 0.809i$
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: \((2,\ 2548,\ (\ :1/2),\ -0.586 - 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4448060629\)
\(L(\frac12)\) \(\approx\) \(0.4448060629\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.16 - 1.72i)T \)
good3 \( 1 - 0.652T + 3T^{2} \)
5 \( 1 + (-1.14 + 1.98i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 6.34T + 11T^{2} \)
17 \( 1 + (-2.48 + 4.30i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 4.78T + 19T^{2} \)
23 \( 1 + (-1.88 - 3.27i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.86 - 6.69i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.130 - 0.225i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.52 - 4.37i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.101 - 0.176i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.84 + 6.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.05 - 8.76i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.59 - 2.76i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.34 - 12.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 1.75T + 61T^{2} \)
67 \( 1 + 4.94T + 67T^{2} \)
71 \( 1 + (-1.93 - 3.35i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.32 + 9.21i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.94 - 12.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.589T + 83T^{2} \)
89 \( 1 + (-1.98 - 3.43i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.13 + 5.43i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.295712994447243143846804016948, −8.436493836054240604289444555698, −7.63214181492084070653746114249, −7.19205345788739433846088035464, −5.65332922451413178572271377114, −5.31685111909294541933395335725, −4.73806776349785007559886682841, −3.13327373024571252527116246187, −2.69860935257819154703465360667, −1.40531879773131658695050530817, 0.12876169783500276359488752841, 2.14362746898206890012162975356, 2.78546741743316626033625531278, 3.39166213523517920380746060590, 4.86732009248702564473367964326, 5.58047764877672797993867365431, 6.17353304338902887168943044711, 7.31370513960660838106458843190, 7.891496418114449198001348209618, 8.402893398555833585042691126457

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