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Properties

Label 2-2548-2548.1247-c0-0-1
Degree $2$
Conductor $2548$
Sign $-0.672 + 0.740i$
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  + (0.900 − 0.433i)2-s + (0.623 − 0.781i)4-s + (−0.623 − 0.781i)7-s + (0.222 − 0.974i)8-s + (−0.900 − 0.433i)9-s + (−0.400 + 0.193i)11-s + (−0.900 + 0.433i)13-s + (−0.900 − 0.433i)14-s + (−0.222 − 0.974i)16-s + (−1.12 − 1.40i)17-s − 18-s + 1.80·19-s + (−0.277 + 0.347i)22-s + (−0.900 − 0.433i)25-s + (−0.623 + 0.781i)26-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)2-s + (0.623 − 0.781i)4-s + (−0.623 − 0.781i)7-s + (0.222 − 0.974i)8-s + (−0.900 − 0.433i)9-s + (−0.400 + 0.193i)11-s + (−0.900 + 0.433i)13-s + (−0.900 − 0.433i)14-s + (−0.222 − 0.974i)16-s + (−1.12 − 1.40i)17-s − 18-s + 1.80·19-s + (−0.277 + 0.347i)22-s + (−0.900 − 0.433i)25-s + (−0.623 + 0.781i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.672 + 0.740i$
Analytic conductor: \(1.27161\)
Root analytic conductor: \(1.12766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (1247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :0),\ -0.672 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.485737757\)
\(L(\frac12)\) \(\approx\) \(1.485737757\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + (0.623 + 0.781i)T \)
13 \( 1 + (0.900 - 0.433i)T \)
good3 \( 1 + (0.900 + 0.433i)T^{2} \)
5 \( 1 + (0.900 + 0.433i)T^{2} \)
11 \( 1 + (0.400 - 0.193i)T + (0.623 - 0.781i)T^{2} \)
17 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
19 \( 1 - 1.80T + T^{2} \)
23 \( 1 + (0.222 + 0.974i)T^{2} \)
29 \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \)
31 \( 1 - 1.80T + T^{2} \)
37 \( 1 + (0.222 - 0.974i)T^{2} \)
41 \( 1 + (0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (0.400 - 0.193i)T + (0.623 - 0.781i)T^{2} \)
53 \( 1 + (-1.24 + 1.56i)T + (-0.222 - 0.974i)T^{2} \)
59 \( 1 + (-0.277 - 1.21i)T + (-0.900 + 0.433i)T^{2} \)
61 \( 1 + (-1.24 - 1.56i)T + (-0.222 + 0.974i)T^{2} \)
67 \( 1 + 1.24T + T^{2} \)
71 \( 1 + (-1.12 + 1.40i)T + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (-0.623 - 0.781i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-1.12 - 0.541i)T + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.623 - 0.781i)T^{2} \)
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

−9.105823027788295442831961995241, −7.78253146264065212536687720453, −7.06388275688582020958727767788, −6.49135309839106829059166269574, −5.51562109689067320843161640281, −4.80906004741559653937388688233, −3.97879165498717365182383615813, −2.97426073057906544465410996610, −2.41931167042540816379458576662, −0.67044699753979397466437076964, 2.18880368392524672712531561135, 2.87301485444655998608868861080, 3.67708352034056773341596045488, 4.89688423584004657644331879393, 5.49412307326128053348708477542, 6.08108591429390892851956915838, 6.90673785658735970162305177676, 7.88648949675556996931143983893, 8.321653720880979180673476450404, 9.233082583625221117487967172961

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