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Properties

Label 2-1456-13.10-c1-0-14
Degree $2$
Conductor $1456$
Sign $0.449 + 0.893i$
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  + (−1.64 − 2.85i)3-s + 0.585i·5-s + (0.866 + 0.5i)7-s + (−3.94 + 6.82i)9-s + (2.18 − 1.26i)11-s + (1.94 − 3.03i)13-s + (1.67 − 0.965i)15-s + (−3.75 + 6.50i)17-s + (1.63 + 0.943i)19-s − 3.29i·21-s + (1.62 + 2.81i)23-s + 4.65·25-s + 16.1·27-s + (3.51 + 6.08i)29-s − 9.37i·31-s + ⋯
L(s)  = 1  + (−0.952 − 1.64i)3-s + 0.261i·5-s + (0.327 + 0.188i)7-s + (−1.31 + 2.27i)9-s + (0.660 − 0.381i)11-s + (0.538 − 0.842i)13-s + (0.431 − 0.249i)15-s + (−0.910 + 1.57i)17-s + (0.375 + 0.216i)19-s − 0.719i·21-s + (0.338 + 0.586i)23-s + 0.931·25-s + 3.10·27-s + (0.652 + 1.12i)29-s − 1.68i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.449 + 0.893i$
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: \((2,\ 1456,\ (\ :1/2),\ 0.449 + 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.257060748\)
\(L(\frac12)\) \(\approx\) \(1.257060748\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.94 + 3.03i)T \)
good3 \( 1 + (1.64 + 2.85i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.585iT - 5T^{2} \)
11 \( 1 + (-2.18 + 1.26i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.75 - 6.50i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.63 - 0.943i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.62 - 2.81i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.51 - 6.08i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.37iT - 31T^{2} \)
37 \( 1 + (-9.17 + 5.29i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.38 + 2.53i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.82 - 4.88i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.57iT - 47T^{2} \)
53 \( 1 + 1.36T + 53T^{2} \)
59 \( 1 + (3.16 + 1.82i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.38 - 7.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.03 - 0.599i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.22 - 1.28i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.75iT - 73T^{2} \)
79 \( 1 - 3.16T + 79T^{2} \)
83 \( 1 + 9.05iT - 83T^{2} \)
89 \( 1 + (-14.9 + 8.60i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.57 + 1.48i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.166423562953301307052335493584, −8.259330406345801684156859809543, −7.72941831330878644253527046356, −6.83169553358688673074755242525, −6.04961950001848150132238369522, −5.73822422176258129890725670650, −4.45055332263624879412691275731, −3.00003811910586677968192969004, −1.80041059940039775319411703190, −0.887479412568323145610330697061, 0.862150932075632900725878735511, 2.85471734904185950756347266477, 4.04875187670425887432726982529, 4.69632629110346527960162239832, 5.10759151637780807958087331911, 6.42053970394822559434069774854, 6.82695267383121201840262726647, 8.400162028204287726506580367847, 9.263854121533453785692726595330, 9.487456285861129025637308118325

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