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Properties

Label 2-2548-13.9-c1-0-23
Degree $2$
Conductor $2548$
Sign $-0.0128 - 0.999i$
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 + 1.73i)3-s + 3·5-s + (−0.499 − 0.866i)9-s + (3.5 + 0.866i)13-s + (−3 + 5.19i)15-s + (1.5 + 2.59i)17-s + (1 + 1.73i)19-s + (3 − 5.19i)23-s + 4·25-s − 4.00·27-s + (−4.5 + 7.79i)29-s − 2·31-s + (3.5 − 6.06i)37-s + (−5 + 5.19i)39-s + (1.5 − 2.59i)41-s + ⋯
L(s)  = 1  + (−0.577 + 0.999i)3-s + 1.34·5-s + (−0.166 − 0.288i)9-s + (0.970 + 0.240i)13-s + (−0.774 + 1.34i)15-s + (0.363 + 0.630i)17-s + (0.229 + 0.397i)19-s + (0.625 − 1.08i)23-s + 0.800·25-s − 0.769·27-s + (−0.835 + 1.44i)29-s − 0.359·31-s + (0.575 − 0.996i)37-s + (−0.800 + 0.832i)39-s + (0.234 − 0.405i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.0128 - 0.999i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (1569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ -0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.050427531\)
\(L(\frac12)\) \(\approx\) \(2.050427531\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 - 0.866i)T \)
good3 \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 3T + 5T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7 - 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.159276383057994106209794015034, −8.701404210543971875957535649090, −7.49061486538593045056987357501, −6.54092737469214894677571665469, −5.65693277118730231406544366721, −5.48914693798515754648559739643, −4.35360554164755668455429029549, −3.61632590612724804454606206656, −2.34246036283960397161040488470, −1.27578609466580240970922668078, 0.824369586940938992834551792064, 1.65839233952873257993634727393, 2.63257431781570951634369926147, 3.80485530209872044572151149795, 5.15518775872213015829032989346, 5.81690629438495181990921708158, 6.22359934995387772023318435337, 7.12493465168629030803883582341, 7.70289016145815184129722390463, 8.785598921257712228847376539179

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