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Properties

Label 2-1456-13.9-c1-0-11
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  + (−1.30 + 2.26i)3-s + 2.61·5-s + (0.5 + 0.866i)7-s + (−1.92 − 3.33i)9-s + (0.927 − 1.60i)11-s + (−2.5 + 2.59i)13-s + (−3.42 + 5.93i)15-s + (0.736 + 1.27i)17-s + (−0.927 − 1.60i)19-s − 2.61·21-s + (−2.23 + 3.87i)23-s + 1.85·25-s + 2.23·27-s + (−3.54 + 6.14i)29-s + 4.70·31-s + ⋯
L(s)  = 1  + (−0.755 + 1.30i)3-s + 1.17·5-s + (0.188 + 0.327i)7-s + (−0.642 − 1.11i)9-s + (0.279 − 0.484i)11-s + (−0.693 + 0.720i)13-s + (−0.884 + 1.53i)15-s + (0.178 + 0.309i)17-s + (−0.212 − 0.368i)19-s − 0.571·21-s + (−0.466 + 0.807i)23-s + 0.370·25-s + 0.430·27-s + (−0.658 + 1.14i)29-s + 0.845·31-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

\(L(1)\) \(\approx\) \(1.263985584\)
\(L(\frac12)\) \(\approx\) \(1.263985584\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.30 - 2.26i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 2.61T + 5T^{2} \)
11 \( 1 + (-0.927 + 1.60i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.736 - 1.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.927 + 1.60i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.23 - 3.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.54 - 6.14i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.70T + 31T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.381 - 0.661i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.28 - 10.8i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.23T + 47T^{2} \)
53 \( 1 - 3.76T + 53T^{2} \)
59 \( 1 + (1.11 + 1.93i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.35 - 11.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.09 + 12.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 6.70T + 83T^{2} \)
89 \( 1 + (2.45 - 4.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.42 + 16.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.859425377401160728046550534586, −9.318275086621025668350574507435, −8.634197935783170691679017579432, −7.30088849876870980559424733201, −6.17312679735306930656083857962, −5.71860605490438164010844193840, −4.90541504820762970113688005184, −4.14715661628870675886569742713, −2.92693113168608544465786712280, −1.62271613302544210424979023036, 0.54377516870769913214941928173, 1.79674521092784992695523399081, 2.49150510071916518720464521296, 4.21317027481963991441209650743, 5.42490829108155258210438819864, 5.88355651269250119452238942075, 6.72899878726561293060031101010, 7.39906002744065998852559987352, 8.124627866772543671562592603399, 9.280981320319036414466507034991

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