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Properties

Label 2-2548-91.88-c1-0-13
Degree $2$
Conductor $2548$
Sign $0.760 - 0.649i$
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  − 0.901·3-s + (1.46 − 0.844i)5-s − 2.18·9-s + 1.24i·11-s + (−3.32 − 1.40i)13-s + (−1.31 + 0.760i)15-s + (−1.80 − 3.12i)17-s + 7.44i·19-s + (4.18 − 7.25i)23-s + (−1.07 + 1.86i)25-s + 4.67·27-s + (3.09 + 5.35i)29-s + (−1.13 − 0.655i)31-s − 1.11i·33-s + (8.90 + 5.14i)37-s + ⋯
L(s)  = 1  − 0.520·3-s + (0.654 − 0.377i)5-s − 0.729·9-s + 0.373i·11-s + (−0.921 − 0.389i)13-s + (−0.340 + 0.196i)15-s + (−0.437 − 0.757i)17-s + 1.70i·19-s + (0.872 − 1.51i)23-s + (−0.214 + 0.372i)25-s + 0.899·27-s + (0.574 + 0.994i)29-s + (−0.203 − 0.117i)31-s − 0.194i·33-s + (1.46 + 0.845i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.760 - 0.649i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ 0.760 - 0.649i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.258502614\)
\(L(\frac12)\) \(\approx\) \(1.258502614\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.32 + 1.40i)T \)
good3 \( 1 + 0.901T + 3T^{2} \)
5 \( 1 + (-1.46 + 0.844i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 1.24iT - 11T^{2} \)
17 \( 1 + (1.80 + 3.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 7.44iT - 19T^{2} \)
23 \( 1 + (-4.18 + 7.25i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.09 - 5.35i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.13 + 0.655i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.90 - 5.14i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.40 - 0.812i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.78 - 8.27i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-10.3 + 5.97i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.95 - 8.58i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.11 + 2.37i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 12.3T + 61T^{2} \)
67 \( 1 - 5.42iT - 67T^{2} \)
71 \( 1 + (-3.90 - 2.25i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.42 + 2.55i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.11 - 5.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 17.4iT - 83T^{2} \)
89 \( 1 + (5.07 + 2.93i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.64 - 0.951i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.969157224774811069120805373747, −8.322825023522794291535274652566, −7.40884609262342196334377083620, −6.55295182501387405612766970279, −5.82614393490616358784177228860, −5.10813196282670164359220266303, −4.51315911718093433981208581889, −3.10254258137585641821120829136, −2.27532751191579185052459243839, −0.947159667722210059242319450477, 0.54377940051548463032725867088, 2.17686662670355716784821440369, 2.83431825349010087568074513138, 4.07448338915619778812463795920, 5.09439531893051857276064343183, 5.67566781515634294331492762857, 6.50331067794407315840301373575, 7.06875860719811804537124638964, 8.063231770020663829599478506158, 8.966856800772966025832830045556

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