↑

Properties

Label 2-731-17.16-c1-0-40
Degree $2$
Conductor $731$
Sign $0.657 - 0.753i$
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  + 2.36·2-s + 1.07i·3-s + 3.60·4-s + 1.87i·5-s + 2.55i·6-s − 1.99i·7-s + 3.79·8-s + 1.83·9-s + 4.44i·10-s + 1.50i·11-s + 3.88i·12-s + 2.58·13-s − 4.71i·14-s − 2.02·15-s + 1.78·16-s + (−2.71 + 3.10i)17-s + ⋯
L(s)  = 1  + 1.67·2-s + 0.622i·3-s + 1.80·4-s + 0.838i·5-s + 1.04i·6-s − 0.752i·7-s + 1.34·8-s + 0.613·9-s + 1.40i·10-s + 0.453i·11-s + 1.12i·12-s + 0.717·13-s − 1.25i·14-s − 0.521·15-s + 0.445·16-s + (−0.657 + 0.753i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.657 - 0.753i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (560, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.657 - 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.62487 + 1.64800i\)
\(L(\frac12)\) \(\approx\) \(3.62487 + 1.64800i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (2.71 - 3.10i)T \)
43 \( 1 + T \)
good2 \( 1 - 2.36T + 2T^{2} \)
3 \( 1 - 1.07iT - 3T^{2} \)
5 \( 1 - 1.87iT - 5T^{2} \)
7 \( 1 + 1.99iT - 7T^{2} \)
11 \( 1 - 1.50iT - 11T^{2} \)
13 \( 1 - 2.58T + 13T^{2} \)
19 \( 1 + 3.80T + 19T^{2} \)
23 \( 1 + 6.00iT - 23T^{2} \)
29 \( 1 + 2.77iT - 29T^{2} \)
31 \( 1 + 2.59iT - 31T^{2} \)
37 \( 1 + 1.88iT - 37T^{2} \)
41 \( 1 - 0.887iT - 41T^{2} \)
47 \( 1 - 3.04T + 47T^{2} \)
53 \( 1 - 0.351T + 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 + 8.67iT - 61T^{2} \)
67 \( 1 + 1.78T + 67T^{2} \)
71 \( 1 + 3.18iT - 71T^{2} \)
73 \( 1 - 1.04iT - 73T^{2} \)
79 \( 1 + 9.83iT - 79T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 - 7.47T + 89T^{2} \)
97 \( 1 + 9.86iT - 97T^{2} \)
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

−10.72707614961009822340103808879, −10.15292892551191866353220373802, −8.817490843092512674410200919403, −7.45867267007741548443648418135, −6.63994806236210437997885839490, −6.08616202961162754734327970882, −4.61689678218094538648439942269, −4.22189342048184686797593744086, −3.36052774691072093649383123441, −2.10779734920946175429487569561, 1.50536521341892397612088792389, 2.72884433880690176396606385717, 3.94796549670037731505059368501, 4.84936389348885320797688563092, 5.65644071195596428953990517764, 6.46529041956275331765402179990, 7.30246509443092818725802982503, 8.526683757590352180200932954046, 9.224108988673972567401090181998, 10.68321204244653999011806495549

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