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Properties

Label 2-2548-7.2-c1-0-25
Degree $2$
Conductor $2548$
Sign $0.386 + 0.922i$
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)5-s + (1.5 − 2.59i)9-s + (1 + 1.73i)11-s − 13-s + (−3 − 5.19i)17-s + (3 − 5.19i)19-s + (−4 + 6.92i)23-s + (0.500 + 0.866i)25-s + 2·29-s + (−5 − 8.66i)31-s + (3 − 5.19i)37-s − 6·41-s + 4·43-s + (3 + 5.19i)45-s + (1 − 1.73i)47-s + ⋯
L(s)  = 1  + (−0.447 + 0.774i)5-s + (0.5 − 0.866i)9-s + (0.301 + 0.522i)11-s − 0.277·13-s + (−0.727 − 1.26i)17-s + (0.688 − 1.19i)19-s + (−0.834 + 1.44i)23-s + (0.100 + 0.173i)25-s + 0.371·29-s + (−0.898 − 1.55i)31-s + (0.493 − 0.854i)37-s − 0.937·41-s + 0.609·43-s + (0.447 + 0.774i)45-s + (0.145 − 0.252i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (1353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.278250461\)
\(L(\frac12)\) \(\approx\) \(1.278250461\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good3 \( 1 + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (5 + 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3 + 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5 - 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−9.040969491223379481320719380049, −7.63656342102172153565445716620, −7.25368151037490098794554295022, −6.70224677680748236299205269460, −5.68903555459744579793984496929, −4.69559890433362762840730282490, −3.86171020252907736804330476234, −3.07124704419659356778672779706, −2.01533046223716986893157225989, −0.46355845509049415574572055847, 1.17317768737656065340148594055, 2.17926442020241845369320320993, 3.52552853922392142885805388737, 4.34255421129253938789531488999, 4.96507792699715729739111225875, 5.94962541115913824875881112916, 6.72177742441423383515781299593, 7.71196150938442075257121876618, 8.414782190775123716792812197500, 8.685391360335856963348159839855

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