↑

Properties

Label 2-1456-13.9-c1-0-21
Degree $2$
Conductor $1456$
Sign $0.872 + 0.488i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.895 − 1.55i)3-s − 3.79·5-s + (0.5 + 0.866i)7-s + (−0.104 − 0.180i)9-s + (−1.89 + 3.28i)11-s + (2.5 − 2.59i)13-s + (−3.39 + 5.88i)15-s + (−1.5 − 2.59i)17-s + (3.68 + 6.38i)19-s + 1.79·21-s + (3 − 5.19i)23-s + 9.37·25-s + 5·27-s + (1.10 − 1.91i)29-s + 31-s + ⋯
L(s)  = 1  + (0.517 − 0.895i)3-s − 1.69·5-s + (0.188 + 0.327i)7-s + (−0.0347 − 0.0602i)9-s + (−0.571 + 0.989i)11-s + (0.693 − 0.720i)13-s + (−0.876 + 1.51i)15-s + (−0.363 − 0.630i)17-s + (0.845 + 1.46i)19-s + 0.390·21-s + (0.625 − 1.08i)23-s + 1.87·25-s + 0.962·27-s + (0.205 − 0.355i)29-s + 0.179·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.872 + 0.488i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ 0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.487979147\)
\(L(\frac12)\) \(\approx\) \(1.487979147\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-2.5 + 2.59i)T \)
good3 \( 1 + (-0.895 + 1.55i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 3.79T + 5T^{2} \)
11 \( 1 + (1.89 - 3.28i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.68 - 6.38i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.10 + 1.91i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.791 + 1.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.18 - 3.78i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 + 10.5T + 53T^{2} \)
59 \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.79 - 6.56i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 9T + 83T^{2} \)
89 \( 1 + (-2.68 + 4.65i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.18 - 15.9i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.207739402442610288282009387927, −8.255811346967625413877049169955, −7.81514773911886468673735015967, −7.41502657550980841875418397703, −6.45075326435481122992972421479, −5.14999852668359935588580303252, −4.30779810780428982738080976302, −3.26720441254734298822387552927, −2.32858832415165138960584423145, −0.875931642096157387873745968288, 0.857379439828470080387499260349, 3.00537190925123896331740241205, 3.58425852346636987486351369163, 4.32248111261051942900417286485, 5.06271732227206335407274748681, 6.45994237457512346858996973262, 7.38161900989075428958462457125, 8.049302885590886165684829300250, 8.895286104388217456773146485892, 9.255750694208036709341047121760

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