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Properties

Label 2-2548-2548.1871-c0-0-0
Degree $2$
Conductor $2548$
Sign $-0.180 - 0.983i$
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  + (−0.733 − 0.680i)2-s + (0.0747 + 0.997i)4-s + (0.0747 − 0.997i)7-s + (0.623 − 0.781i)8-s + (0.955 + 0.294i)9-s + (−1.88 + 0.582i)11-s + (−0.222 + 0.974i)13-s + (−0.733 + 0.680i)14-s + (−0.988 + 0.149i)16-s + (−1.21 − 0.825i)17-s + (−0.5 − 0.866i)18-s + (−0.955 + 1.65i)19-s + (1.78 + 0.858i)22-s + (−0.733 + 0.680i)25-s + (0.826 − 0.563i)26-s + ⋯
L(s)  = 1  + (−0.733 − 0.680i)2-s + (0.0747 + 0.997i)4-s + (0.0747 − 0.997i)7-s + (0.623 − 0.781i)8-s + (0.955 + 0.294i)9-s + (−1.88 + 0.582i)11-s + (−0.222 + 0.974i)13-s + (−0.733 + 0.680i)14-s + (−0.988 + 0.149i)16-s + (−1.21 − 0.825i)17-s + (−0.5 − 0.866i)18-s + (−0.955 + 1.65i)19-s + (1.78 + 0.858i)22-s + (−0.733 + 0.680i)25-s + (0.826 − 0.563i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.180 - 0.983i$
Analytic conductor: \(1.27161\)
Root analytic conductor: \(1.12766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (1871, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :0),\ -0.180 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2835923326\)
\(L(\frac12)\) \(\approx\) \(0.2835923326\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.733 + 0.680i)T \)
7 \( 1 + (-0.0747 + 0.997i)T \)
13 \( 1 + (0.222 - 0.974i)T \)
good3 \( 1 + (-0.955 - 0.294i)T^{2} \)
5 \( 1 + (0.733 - 0.680i)T^{2} \)
11 \( 1 + (1.88 - 0.582i)T + (0.826 - 0.563i)T^{2} \)
17 \( 1 + (1.21 + 0.825i)T + (0.365 + 0.930i)T^{2} \)
19 \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.365 + 0.930i)T^{2} \)
29 \( 1 + (0.658 - 0.317i)T + (0.623 - 0.781i)T^{2} \)
31 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.988 + 0.149i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.535 + 0.496i)T + (0.0747 + 0.997i)T^{2} \)
53 \( 1 + (0.0747 + 0.997i)T + (-0.988 + 0.149i)T^{2} \)
59 \( 1 + (-0.0546 + 0.139i)T + (-0.733 - 0.680i)T^{2} \)
61 \( 1 + (0.0747 - 0.997i)T + (-0.988 - 0.149i)T^{2} \)
67 \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-1.32 - 0.636i)T + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.0747 + 0.997i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.826 - 0.563i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.684094622924694167080990462047, −8.474565775468683319090672534210, −7.78546991307587525630486906629, −7.25947085125881084686670975940, −6.63915713260458660493739484750, −5.08211354189710708476277405949, −4.37811793762004732649237012308, −3.66449578541814669421351575061, −2.29325543022362552310906082506, −1.70056708306841474766495396752, 0.21748648894908037927219660711, 2.07724073271294033472727147235, 2.73390358083124848397350728028, 4.41038421930386521771569619662, 5.11866660215894559082217377583, 5.93924680528601597126158403588, 6.52938505996657220485482117449, 7.56561537391982230224827380540, 8.109964687865344039912799006781, 8.744409862051327495291265649183

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