Properties

Label 2-1456-13.10-c1-0-4
Degree $2$
Conductor $1456$
Sign $-0.848 - 0.529i$
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  + (1.33 + 2.30i)3-s − 3.16i·5-s + (−0.866 − 0.5i)7-s + (−2.03 + 3.53i)9-s + (−5.14 + 2.97i)11-s + (−0.0766 + 3.60i)13-s + (7.28 − 4.20i)15-s + (−1.34 + 2.33i)17-s + (−1.69 − 0.978i)19-s − 2.66i·21-s + (1.36 + 2.36i)23-s − 4.99·25-s − 2.86·27-s + (2.99 + 5.19i)29-s + 1.15i·31-s + ⋯
L(s)  = 1  + (0.767 + 1.33i)3-s − 1.41i·5-s + (−0.327 − 0.188i)7-s + (−0.679 + 1.17i)9-s + (−1.55 + 0.895i)11-s + (−0.0212 + 0.999i)13-s + (1.88 − 1.08i)15-s + (−0.327 + 0.567i)17-s + (−0.388 − 0.224i)19-s − 0.580i·21-s + (0.284 + 0.492i)23-s − 0.999·25-s − 0.551·27-s + (0.556 + 0.964i)29-s + 0.206i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $-0.848 - 0.529i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ -0.848 - 0.529i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.147451246\)
\(L(\frac12)\) \(\approx\) \(1.147451246\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0766 - 3.60i)T \)
good3 \( 1 + (-1.33 - 2.30i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 3.16iT - 5T^{2} \)
11 \( 1 + (5.14 - 2.97i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.34 - 2.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.69 + 0.978i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.36 - 2.36i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.99 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.15iT - 31T^{2} \)
37 \( 1 + (5.63 - 3.25i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.23 - 1.86i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.49 - 6.05i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.456iT - 47T^{2} \)
53 \( 1 - 0.399T + 53T^{2} \)
59 \( 1 + (4.16 + 2.40i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.578 + 1.00i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.43 + 3.13i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.90 + 2.25i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 8.30iT - 73T^{2} \)
79 \( 1 - 7.91T + 79T^{2} \)
83 \( 1 + 6.19iT - 83T^{2} \)
89 \( 1 + (-3.08 + 1.78i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.96 + 1.71i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.817112646269171695205060942818, −8.986393916857293844587348842506, −8.583537251322333200041246319966, −7.75002832660779215714773218467, −6.61719093355053506507571360442, −5.10924494643204164436581998612, −4.85884280368581702232960899939, −4.06543147359003012759917844834, −3.00447623593290380708445969551, −1.77033382219601657973763286313, 0.38171625029470782758074748190, 2.30637936926682309877175565963, 2.75834331478044198002687550151, 3.47037753593701446030264579544, 5.28123826388745418002015820047, 6.19674206052764417602942979820, 6.86060612027684887012209125062, 7.62153030004322132799323302285, 8.136498901342510492531127580845, 8.898517400359097339825185831452

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