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Properties

Label 2-1456-13.4-c1-0-9
Degree $2$
Conductor $1456$
Sign $0.489 - 0.872i$
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  + (−0.291 + 0.504i)3-s − 1.68i·5-s + (−0.866 + 0.5i)7-s + (1.33 + 2.30i)9-s + (0.315 + 0.182i)11-s + (1.80 + 3.12i)13-s + (0.851 + 0.491i)15-s + (−1.59 − 2.75i)17-s + (−1.25 + 0.721i)19-s − 0.582i·21-s + (2.54 − 4.40i)23-s + 2.15·25-s − 3.29·27-s + (−4.09 + 7.09i)29-s + 4.69i·31-s + ⋯
L(s)  = 1  + (−0.168 + 0.291i)3-s − 0.754i·5-s + (−0.327 + 0.188i)7-s + (0.443 + 0.768i)9-s + (0.0952 + 0.0549i)11-s + (0.499 + 0.866i)13-s + (0.219 + 0.126i)15-s + (−0.386 − 0.669i)17-s + (−0.286 + 0.165i)19-s − 0.127i·21-s + (0.529 − 0.917i)23-s + 0.430·25-s − 0.634·27-s + (−0.761 + 1.31i)29-s + 0.843i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.489 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.489 - 0.872i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ 0.489 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.443840800\)
\(L(\frac12)\) \(\approx\) \(1.443840800\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-1.80 - 3.12i)T \)
good3 \( 1 + (0.291 - 0.504i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.68iT - 5T^{2} \)
11 \( 1 + (-0.315 - 0.182i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.59 + 2.75i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.25 - 0.721i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.54 + 4.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.09 - 7.09i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.69iT - 31T^{2} \)
37 \( 1 + (-5.46 - 3.15i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.04 + 2.91i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.386 - 0.669i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.7iT - 47T^{2} \)
53 \( 1 - 1.37T + 53T^{2} \)
59 \( 1 + (-8.10 + 4.68i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.51 - 7.81i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.6 - 6.73i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.13 + 3.54i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 2.16iT - 73T^{2} \)
79 \( 1 - 6.88T + 79T^{2} \)
83 \( 1 + 0.567iT - 83T^{2} \)
89 \( 1 + (0.986 + 0.569i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.86 + 3.96i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.494446070798544718406479490204, −8.937234748899777243585660805185, −8.238765913703004144699063899888, −7.10960095326458898879432446482, −6.48031432367857031904880653207, −5.24505808283747081288006648100, −4.74139342650271068151717387015, −3.82531102963144305087402869742, −2.48381447161580878849033899631, −1.24453201947198164410251868041, 0.66567053017726681481662773492, 2.16299073063294698636216958086, 3.41910141972432898384396565164, 3.99639622580362013997099246264, 5.44049996788862542773304773497, 6.27155973516686243593449189049, 6.84173600769625610259136862184, 7.65540738885515502181018402886, 8.527323834645394598509367718827, 9.552702270136097936375980756012

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