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Properties

Label 2-1456-7.4-c1-0-28
Degree $2$
Conductor $1456$
Sign $0.605 + 0.795i$
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  + (−1.11 + 1.93i)3-s + (−1.11 − 1.93i)5-s + (2 − 1.73i)7-s + (−1 − 1.73i)9-s + (−1.5 + 2.59i)11-s − 13-s + 5.00·15-s + (−0.736 + 1.27i)17-s + (1.5 + 2.59i)19-s + (1.11 + 5.80i)21-s + (−4.11 − 7.13i)23-s − 2.23·27-s + 4.47·29-s + (2.5 − 4.33i)31-s + (−3.35 − 5.80i)33-s + ⋯
L(s)  = 1  + (−0.645 + 1.11i)3-s + (−0.499 − 0.866i)5-s + (0.755 − 0.654i)7-s + (−0.333 − 0.577i)9-s + (−0.452 + 0.783i)11-s − 0.277·13-s + 1.29·15-s + (−0.178 + 0.309i)17-s + (0.344 + 0.596i)19-s + (0.243 + 1.26i)21-s + (−0.858 − 1.48i)23-s − 0.430·27-s + 0.830·29-s + (0.449 − 0.777i)31-s + (−0.583 − 1.01i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9102758321\)
\(L(\frac12)\) \(\approx\) \(0.9102758321\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 1.73i)T \)
13 \( 1 + T \)
good3 \( 1 + (1.11 - 1.93i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.11 + 1.93i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.736 - 1.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.5 - 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.11 + 7.13i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.35 + 4.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.47T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (3.73 + 6.47i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.73 + 6.47i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.736 - 1.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.5 + 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.94T + 71T^{2} \)
73 \( 1 + (-1.35 + 2.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.35 + 2.34i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (1.11 + 1.93i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.41T + 97T^{2} \)
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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

−9.645659539542741424069355104949, −8.484970178916656850622176837070, −8.009229300726425270482260351376, −7.02879157940109015850314563910, −5.83001715591664009080558494370, −4.86267938989088297320249158896, −4.52012560832327671413351123879, −3.83049004625824186962950776871, −2.07964321584152371411486925037, −0.44674647239369460705760521087, 1.17502602896617912762863246050, 2.44383402011297435569639190797, 3.37592547999681302025159735140, 4.84090489106746342224804788090, 5.67361438612894570670925081792, 6.42468479356691552278898378776, 7.27632591972140893493524770532, 7.77793213008911137188629703954, 8.593570213025123543603791360495, 9.641026447313245779597978340294

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