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Properties

Label 2-2548-13.10-c1-0-45
Degree $2$
Conductor $2548$
Sign $-0.999 + 0.0445i$
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 − 2.84i·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 + (−0.806 + 1.39i)17-s + (−4.64 − 2.68i)19-s + (−0.575 − 0.997i)23-s − 3.12·25-s + 3.45·27-s + (−1.88 − 3.26i)29-s + 0.375i·31-s + (−0.959 − 0.553i)33-s + (1.96 − 1.13i)37-s + ⋯
L(s)  = 1  + (0.177 + 0.307i)3-s − 1.27i·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 + (−0.195 + 0.338i)17-s + (−1.06 − 0.615i)19-s + (−0.120 − 0.207i)23-s − 0.624·25-s + 0.665·27-s + (−0.350 − 0.606i)29-s + 0.0674i·31-s + (−0.167 − 0.0964i)33-s + (0.323 − 0.186i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.999 + 0.0445i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (1765, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ -0.999 + 0.0445i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5063814109\)
\(L(\frac12)\) \(\approx\) \(0.5063814109\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.84iT - 5T^{2} \)
11 \( 1 + (1.56 - 0.901i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.806 - 1.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.64 + 2.68i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.575 + 0.997i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.88 + 3.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.375iT - 31T^{2} \)
37 \( 1 + (-1.96 + 1.13i)T + (18.5 - 32.0i)T^{2} \)
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 \( 1 + 1.43iT - 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 + (6.17 + 3.56i)T + (29.5 + 51.0i)T^{2} \)
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 \( 1 - 5.62iT - 73T^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 + 2.93iT - 83T^{2} \)
89 \( 1 + (11.1 - 6.45i)T + (44.5 - 77.0i)T^{2} \)
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

−8.587354715942117627033407535395, −7.969782705783723170630133655976, −6.94622514522256474310267658964, −6.25527259027518917908415376320, −5.11058731927990091309326285753, −4.55971019445478959029391653844, −3.92551466989559295422567334015, −2.60128419648311093041923326479, −1.52308312133860264505106339331, −0.15214416606061303892434218263, 1.82194904155179814759656772272, 2.64127513852399384111346064708, 3.38713459147286134183531898429, 4.57071136041304311410092112649, 5.40051247123334381243631712307, 6.40357218042821430846270997674, 7.01280013743719064969247406341, 7.76802602043628476809668609223, 8.191533432246748508654393364591, 9.376391271666685746976300372379

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