↑

Properties

Label 2-731-731.208-c1-0-5
Degree $2$
Conductor $731$
Sign $0.689 - 0.724i$
Analytic cond. $5.83706$
Root an. cond. $2.41600$
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  − 2.28i·2-s + (1.18 + 0.316i)3-s − 3.23·4-s + (−0.126 − 0.0339i)5-s + (0.723 − 2.70i)6-s + (−4.66 + 1.24i)7-s + 2.81i·8-s + (−1.30 − 0.752i)9-s + (−0.0775 + 0.289i)10-s + (4.02 + 4.02i)11-s + (−3.81 − 1.02i)12-s + (−3.19 + 5.53i)13-s + (2.85 + 10.6i)14-s + (−0.138 − 0.0801i)15-s − 0.0255·16-s + (1.31 + 3.90i)17-s + ⋯
L(s)  = 1  − 1.61i·2-s + (0.681 + 0.182i)3-s − 1.61·4-s + (−0.0566 − 0.0151i)5-s + (0.295 − 1.10i)6-s + (−1.76 + 0.472i)7-s + 0.994i·8-s + (−0.434 − 0.250i)9-s + (−0.0245 + 0.0915i)10-s + (1.21 + 1.21i)11-s + (−1.10 − 0.295i)12-s + (−0.886 + 1.53i)13-s + (0.763 + 2.84i)14-s + (−0.0358 − 0.0206i)15-s − 0.00638·16-s + (0.318 + 0.947i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.689 - 0.724i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.689 - 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.475998 + 0.203975i\)
\(L(\frac12)\) \(\approx\) \(0.475998 + 0.203975i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-1.31 - 3.90i)T \)
43 \( 1 + (2.53 - 6.04i)T \)
good2 \( 1 + 2.28iT - 2T^{2} \)
3 \( 1 + (-1.18 - 0.316i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (0.126 + 0.0339i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (4.66 - 1.24i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-4.02 - 4.02i)T + 11iT^{2} \)
13 \( 1 + (3.19 - 5.53i)T + (-6.5 - 11.2i)T^{2} \)
19 \( 1 + (2.72 - 1.57i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.24 + 4.64i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.01 + 7.53i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-2.23 - 0.599i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (8.74 + 2.34i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-5.31 - 5.31i)T + 41iT^{2} \)
47 \( 1 - 1.76T + 47T^{2} \)
53 \( 1 + (-1.36 + 0.790i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.73iT - 59T^{2} \)
61 \( 1 + (-2.45 + 0.657i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.59 + 2.75i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.911 - 3.40i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.65 - 6.16i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (10.3 - 2.77i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (1.91 - 1.10i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.42 + 4.20i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.3 - 10.3i)T - 97iT^{2} \)
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

−10.20060859181938083856595871597, −9.614955179968677491863576561503, −9.349591705578405441498458814243, −8.478226521259544431439354496197, −6.84514568750103544859655248021, −6.19611075932457087370252073246, −4.21977534109590492941214722080, −3.91335207654955905617687141401, −2.69167160964109040063258383660, −1.97219055409067648363375570370, 0.22974000464388904790693184134, 3.00932915596167550170066659288, 3.64606836214594056920552935336, 5.35447506729560081893300496237, 5.91484799809144298998187861862, 7.03467261037669209155998782584, 7.36397878774799150823637888479, 8.497174997824464063040970549733, 9.092050308676845002367019682655, 9.803756620529675360943989064801

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