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Properties

Label 2-1456-13.10-c1-0-0
Degree $2$
Conductor $1456$
Sign $-0.872 + 0.489i$
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  + (0.439 + 0.761i)3-s + 3.76i·5-s + (−0.866 − 0.5i)7-s + (1.11 − 1.92i)9-s + (−2.30 + 1.33i)11-s + (−3.12 − 1.80i)13-s + (−2.87 + 1.65i)15-s + (−2.54 + 4.40i)17-s + (−6.23 − 3.60i)19-s − 0.879i·21-s + (−2.28 − 3.95i)23-s − 9.19·25-s + 4.59·27-s + (3.57 + 6.19i)29-s − 4.83i·31-s + ⋯
L(s)  = 1  + (0.253 + 0.439i)3-s + 1.68i·5-s + (−0.327 − 0.188i)7-s + (0.371 − 0.642i)9-s + (−0.695 + 0.401i)11-s + (−0.866 − 0.499i)13-s + (−0.741 + 0.427i)15-s + (−0.616 + 1.06i)17-s + (−1.43 − 0.826i)19-s − 0.191i·21-s + (−0.476 − 0.824i)23-s − 1.83·25-s + 0.884·27-s + (0.663 + 1.14i)29-s − 0.868i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $-0.872 + 0.489i$
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: \((2,\ 1456,\ (\ :1/2),\ -0.872 + 0.489i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4176092532\)
\(L(\frac12)\) \(\approx\) \(0.4176092532\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.12 + 1.80i)T \)
good3 \( 1 + (-0.439 - 0.761i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3.76iT - 5T^{2} \)
11 \( 1 + (2.30 - 1.33i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.54 - 4.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.23 + 3.60i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.28 + 3.95i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.57 - 6.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.83iT - 31T^{2} \)
37 \( 1 + (-5.38 + 3.10i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (10.2 - 5.92i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.870 - 1.50i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.21iT - 47T^{2} \)
53 \( 1 - 6.77T + 53T^{2} \)
59 \( 1 + (6.68 + 3.86i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.12 - 1.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.79 + 1.03i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.37 + 4.83i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 8.61iT - 73T^{2} \)
79 \( 1 - 10.6T + 79T^{2} \)
83 \( 1 - 14.9iT - 83T^{2} \)
89 \( 1 + (0.0725 - 0.0419i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.76 + 5.64i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.24839207965662936547216227743, −9.376689892147320123733423755788, −8.338278762066993015217388094945, −7.47242926525366413110950572198, −6.57458313685626716639848430458, −6.32541369908983687290165367962, −4.76688400383808355767464793524, −3.94361061377895326511098661669, −2.94954548257169127619583576471, −2.28489325037294715215117946140, 0.14764558450673324199627511630, 1.67437830919185873259077326876, 2.54040680759841533910898804432, 4.14241689650218488956356463964, 4.85590778190662836043488720709, 5.54713327235842932625653241650, 6.67999796739125909689020879792, 7.62650381059138361998522086791, 8.354828379647136102983654727639, 8.835344239944422329511976135769

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