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Properties

Label 2-2548-2548.1507-c0-0-1
Degree $2$
Conductor $2548$
Sign $-0.304 + 0.952i$
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  + (0.826 − 0.563i)2-s + (0.365 − 0.930i)4-s + (0.365 + 0.930i)7-s + (−0.222 − 0.974i)8-s + (0.0747 − 0.997i)9-s + (−0.109 − 1.46i)11-s + (−0.900 − 0.433i)13-s + (0.826 + 0.563i)14-s + (−0.733 − 0.680i)16-s + (−1.63 − 0.246i)17-s + (−0.5 − 0.866i)18-s + (−0.0747 + 0.129i)19-s + (−0.914 − 1.14i)22-s + (0.826 + 0.563i)25-s + (−0.988 + 0.149i)26-s + ⋯
L(s)  = 1  + (0.826 − 0.563i)2-s + (0.365 − 0.930i)4-s + (0.365 + 0.930i)7-s + (−0.222 − 0.974i)8-s + (0.0747 − 0.997i)9-s + (−0.109 − 1.46i)11-s + (−0.900 − 0.433i)13-s + (0.826 + 0.563i)14-s + (−0.733 − 0.680i)16-s + (−1.63 − 0.246i)17-s + (−0.5 − 0.866i)18-s + (−0.0747 + 0.129i)19-s + (−0.914 − 1.14i)22-s + (0.826 + 0.563i)25-s + (−0.988 + 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.304 + 0.952i$
Analytic conductor: \(1.27161\)
Root analytic conductor: \(1.12766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (1507, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :0),\ -0.304 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.805608438\)
\(L(\frac12)\) \(\approx\) \(1.805608438\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.826 + 0.563i)T \)
7 \( 1 + (-0.365 - 0.930i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
good3 \( 1 + (-0.0747 + 0.997i)T^{2} \)
5 \( 1 + (-0.826 - 0.563i)T^{2} \)
11 \( 1 + (0.109 + 1.46i)T + (-0.988 + 0.149i)T^{2} \)
17 \( 1 + (1.63 + 0.246i)T + (0.955 + 0.294i)T^{2} \)
19 \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.955 + 0.294i)T^{2} \)
29 \( 1 + (-1.19 + 1.49i)T + (-0.222 - 0.974i)T^{2} \)
31 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.733 - 0.680i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (-1.57 + 1.07i)T + (0.365 - 0.930i)T^{2} \)
53 \( 1 + (0.365 - 0.930i)T + (-0.733 - 0.680i)T^{2} \)
59 \( 1 + (-0.698 + 0.215i)T + (0.826 - 0.563i)T^{2} \)
61 \( 1 + (0.365 + 0.930i)T + (-0.733 + 0.680i)T^{2} \)
67 \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-1.03 - 1.29i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.365 - 0.930i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (0.988 + 0.149i)T^{2} \)
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

−8.825308168127772186372812203347, −8.403932984561601355059837127270, −6.97721115096453849810479962967, −6.39109029100605288299766452409, −5.62097103889545794188387410877, −4.91696918094991722771624761997, −4.02870484551053867130170058221, −2.93425621707910502329410842077, −2.46457111160381990985812761737, −0.900459524664371969011678514451, 1.98553367067005387598552693473, 2.63556677305568919865147736302, 4.20367580607025986474439214197, 4.60638695759080872027746352034, 5.03447121938452896690313881645, 6.46474030177280384342597396615, 6.97530076446518506643058897335, 7.56743630979825643593403302146, 8.252967929520591750764922958546, 9.175822589085429128347522025918

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