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Properties

Label 2-1456-13.9-c1-0-29
Degree $2$
Conductor $1456$
Sign $0.872 + 0.488i$
Analytic cond. $11.6262$
Root an. cond. $3.40972$
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  + (−1.39 + 2.41i)3-s + 0.791·5-s + (0.5 + 0.866i)7-s + (−2.39 − 4.14i)9-s + (0.395 − 0.685i)11-s + (2.5 − 2.59i)13-s + (−1.10 + 1.91i)15-s + (−1.5 − 2.59i)17-s + (−3.18 − 5.51i)19-s − 2.79·21-s + (3 − 5.19i)23-s − 4.37·25-s + 5.00·27-s + (3.39 − 5.88i)29-s + 31-s + ⋯
L(s)  = 1  + (−0.805 + 1.39i)3-s + 0.353·5-s + (0.188 + 0.327i)7-s + (−0.798 − 1.38i)9-s + (0.119 − 0.206i)11-s + (0.693 − 0.720i)13-s + (−0.285 + 0.493i)15-s + (−0.363 − 0.630i)17-s + (−0.731 − 1.26i)19-s − 0.609·21-s + (0.625 − 1.08i)23-s − 0.874·25-s + 0.962·27-s + (0.630 − 1.09i)29-s + 0.179·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.872 + 0.488i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ 0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9811879545\)
\(L(\frac12)\) \(\approx\) \(0.9811879545\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-2.5 + 2.59i)T \)
good3 \( 1 + (1.39 - 2.41i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 0.791T + 5T^{2} \)
11 \( 1 + (-0.395 + 0.685i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.18 + 5.51i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.39 + 5.88i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.79 - 6.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.68 + 8.11i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.16T + 47T^{2} \)
53 \( 1 + 1.41T + 53T^{2} \)
59 \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.791 + 1.37i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 9T + 83T^{2} \)
89 \( 1 + (4.18 - 7.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.31 - 4.00i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.579701450671227332318475731975, −8.858475624429932346994091347854, −8.105794700805319148155899543606, −6.63154637704538821387346480786, −6.08044888859144223704159669311, −5.09656539507235195969907560723, −4.62150195508064702846058875151, −3.57023775443568657500864428077, −2.44596979311537725206089723778, −0.45186471159677713144858379217, 1.38774276412236079553630423974, 1.85773496886157162696354245806, 3.52467310334004283353342728180, 4.70179838041135286856585580517, 5.78801052771645046182105811507, 6.35792715198300544771460203685, 6.98109723128386054223233690505, 7.87287272888664553263603274163, 8.536948894785927362791167718048, 9.618762938724876291110867608109

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