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Properties

Label 2-1456-13.10-c1-0-17
Degree $2$
Conductor $1456$
Sign $0.529 - 0.848i$
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  + (0.911 + 1.57i)3-s + 0.0393i·5-s + (−0.866 − 0.5i)7-s + (−0.162 + 0.281i)9-s + (−0.468 + 0.270i)11-s + (3.60 + 0.0766i)13-s + (−0.0621 + 0.0358i)15-s + (0.944 − 1.63i)17-s + (0.616 + 0.356i)19-s − 1.82i·21-s + (1.97 + 3.42i)23-s + 4.99·25-s + 4.87·27-s + (0.000585 + 0.00101i)29-s + 9.38i·31-s + ⋯
L(s)  = 1  + (0.526 + 0.911i)3-s + 0.0175i·5-s + (−0.327 − 0.188i)7-s + (−0.0542 + 0.0939i)9-s + (−0.141 + 0.0815i)11-s + (0.999 + 0.0212i)13-s + (−0.0160 + 0.00926i)15-s + (0.229 − 0.396i)17-s + (0.141 + 0.0817i)19-s − 0.397i·21-s + (0.412 + 0.714i)23-s + 0.999·25-s + 0.938·27-s + (0.000108 + 0.000188i)29-s + 1.68i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.529 - 0.848i$
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: \((2,\ 1456,\ (\ :1/2),\ 0.529 - 0.848i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.100174945\)
\(L(\frac12)\) \(\approx\) \(2.100174945\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.60 - 0.0766i)T \)
good3 \( 1 + (-0.911 - 1.57i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.0393iT - 5T^{2} \)
11 \( 1 + (0.468 - 0.270i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.944 + 1.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.616 - 0.356i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.97 - 3.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.000585 - 0.00101i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.38iT - 31T^{2} \)
37 \( 1 + (0.411 - 0.237i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.07 - 0.620i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.93 + 3.35i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.91iT - 47T^{2} \)
53 \( 1 + 2.56T + 53T^{2} \)
59 \( 1 + (-4.73 - 2.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.55 + 2.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.67 - 1.54i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.59 + 3.23i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 8.61iT - 73T^{2} \)
79 \( 1 - 4.88T + 79T^{2} \)
83 \( 1 - 1.65iT - 83T^{2} \)
89 \( 1 + (-3.15 + 1.82i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.1 - 7.56i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.498245681585894600875429995382, −9.023920212926140300562440957017, −8.265833002480256677708567159285, −7.20904376451878221149390530318, −6.44162750390979587310260302340, −5.33884824465220609895451412125, −4.49331967333841468608591025235, −3.51012545572672093121244193600, −2.96721995991971052462120974388, −1.23877734292443246279049149130, 0.959380864490128936443444343912, 2.16836562646043262204054960021, 3.08637775976134225613211384690, 4.16829157001633877330370315511, 5.36076317769898431154802995308, 6.31634400129477960764250162259, 6.95398392939736403766732014417, 7.87785157184990570469676613472, 8.469643464740799192670587422359, 9.162045737483169041505785292379

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