Properties

Label 2-1456-13.4-c1-0-36
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: $\chi_{1456} (225, \cdot )$
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

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

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