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Properties

Label 2-2548-91.23-c1-0-20
Degree $2$
Conductor $2548$
Sign $0.719 - 0.694i$
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  + (0.307 − 0.532i)3-s + (2.46 + 1.42i)5-s + (1.31 + 2.27i)9-s + (1.56 + 0.901i)11-s + (−3.01 + 1.97i)13-s + (1.51 − 0.876i)15-s + 1.61·17-s + (4.64 − 2.68i)19-s + 1.15·23-s + (1.56 + 2.70i)25-s + 3.45·27-s + (−1.88 − 3.26i)29-s + (0.325 − 0.187i)31-s + (0.959 − 0.553i)33-s + 2.27i·37-s + ⋯
L(s)  = 1  + (0.177 − 0.307i)3-s + (1.10 + 0.637i)5-s + (0.436 + 0.756i)9-s + (0.470 + 0.271i)11-s + (−0.835 + 0.548i)13-s + (0.391 − 0.226i)15-s + 0.391·17-s + (1.06 − 0.615i)19-s + 0.240·23-s + (0.312 + 0.540i)25-s + 0.665·27-s + (−0.350 − 0.606i)29-s + (0.0584 − 0.0337i)31-s + (0.167 − 0.0964i)33-s + 0.373i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.719 - 0.694i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ 0.719 - 0.694i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.546948497\)
\(L(\frac12)\) \(\approx\) \(2.546948497\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.01 - 1.97i)T \)
good3 \( 1 + (-0.307 + 0.532i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.46 - 1.42i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.56 - 0.901i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 1.61T + 17T^{2} \)
19 \( 1 + (-4.64 + 2.68i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.15T + 23T^{2} \)
29 \( 1 + (1.88 + 3.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.325 + 0.187i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.27iT - 37T^{2} \)
41 \( 1 + (5.59 - 3.23i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.235 - 0.407i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.24 - 0.719i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.06 - 10.5i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 7.12iT - 59T^{2} \)
61 \( 1 + (2.80 + 4.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.40 - 0.811i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.79 - 5.07i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.86 + 2.81i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.18 - 12.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.93iT - 83T^{2} \)
89 \( 1 + 12.9iT - 89T^{2} \)
97 \( 1 + (10.4 + 6.05i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.193687680015572186208583361255, −8.136367226815644278690794697181, −7.22545254777264531919286929904, −6.91219248407361181937166969778, −5.93332648502320316605025044876, −5.13742376767319687516733288640, −4.30861522751734589723509589529, −2.96976187059198606644422712407, −2.26146433957821405486614095161, −1.36591392643068314981597592279, 0.892428120974357509393664848376, 1.87648993963708400000066654022, 3.14143276190054167473235817658, 3.89922439425289707249327276818, 5.14070967448955697937149489512, 5.46017503557747704288521963901, 6.46612124919636417034491243969, 7.22297531162080542910934747592, 8.176555261613127494929694976029, 9.034856867370351094774124764915

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