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Properties

Label 2-2548-2548.1975-c0-0-0
Degree $2$
Conductor $2548$
Sign $-0.761 + 0.648i$
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  + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.900 + 0.433i)7-s + (0.623 − 0.781i)8-s + (−0.222 − 0.974i)9-s + (−0.277 + 1.21i)11-s + (−0.222 + 0.974i)13-s + (−0.222 − 0.974i)14-s + (0.623 + 0.781i)16-s + (0.400 − 0.193i)17-s + 18-s − 0.445·19-s + (−1.12 − 0.541i)22-s + (−0.222 − 0.974i)25-s + (−0.900 − 0.433i)26-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.900 + 0.433i)7-s + (0.623 − 0.781i)8-s + (−0.222 − 0.974i)9-s + (−0.277 + 1.21i)11-s + (−0.222 + 0.974i)13-s + (−0.222 − 0.974i)14-s + (0.623 + 0.781i)16-s + (0.400 − 0.193i)17-s + 18-s − 0.445·19-s + (−1.12 − 0.541i)22-s + (−0.222 − 0.974i)25-s + (−0.900 − 0.433i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.761 + 0.648i$
Analytic conductor: \(1.27161\)
Root analytic conductor: \(1.12766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (1975, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :0),\ -0.761 + 0.648i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2218406147\)
\(L(\frac12)\) \(\approx\) \(0.2218406147\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.222 - 0.974i)T \)
7 \( 1 + (0.900 - 0.433i)T \)
13 \( 1 + (0.222 - 0.974i)T \)
good3 \( 1 + (0.222 + 0.974i)T^{2} \)
5 \( 1 + (0.222 + 0.974i)T^{2} \)
11 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
17 \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \)
19 \( 1 + 0.445T + T^{2} \)
23 \( 1 + (-0.623 - 0.781i)T^{2} \)
29 \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \)
31 \( 1 + 0.445T + T^{2} \)
37 \( 1 + (-0.623 + 0.781i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
53 \( 1 + (1.80 + 0.867i)T + (0.623 + 0.781i)T^{2} \)
59 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
61 \( 1 + (1.80 - 0.867i)T + (0.623 - 0.781i)T^{2} \)
67 \( 1 + 1.80T + T^{2} \)
71 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (0.900 - 0.433i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.900 - 0.433i)T^{2} \)
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

−9.420712872276977990395781650921, −8.954794250233909157658867582212, −7.896241634532518845785044772696, −7.17035030250565119061430040107, −6.46535152100749143087983885134, −5.99981446656014712208061861843, −4.92604876084065933709630126148, −4.17611773901845600939195728131, −3.17939759200446260153946576703, −1.79318805726508741604400361597, 0.15215382553788516007572386100, 1.70657103113020274156470119016, 3.00182115033747969205920532618, 3.34774872814815561739776879039, 4.46756231848982927271263747369, 5.48281603889987292930697715989, 6.03751925282809246673194393041, 7.57491358017076586781980085961, 7.79644565682351171793552349572, 8.807601058038574073776651605736

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