↑

Properties

Label 2-731-17.16-c1-0-12
Degree $2$
Conductor $731$
Sign $-0.927 - 0.374i$
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  − 1.28·2-s + 2.62i·3-s − 0.361·4-s − 1.51i·5-s − 3.35i·6-s + 3.46i·7-s + 3.02·8-s − 3.87·9-s + 1.93i·10-s + 2.66i·11-s − 0.947i·12-s + 6.10·13-s − 4.43i·14-s + 3.97·15-s − 3.14·16-s + (3.82 + 1.54i)17-s + ⋯
L(s)  = 1  − 0.905·2-s + 1.51i·3-s − 0.180·4-s − 0.677i·5-s − 1.37i·6-s + 1.30i·7-s + 1.06·8-s − 1.29·9-s + 0.613i·10-s + 0.802i·11-s − 0.273i·12-s + 1.69·13-s − 1.18i·14-s + 1.02·15-s − 0.786·16-s + (0.927 + 0.374i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.927 - 0.374i$
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: \((2,\ 731,\ (\ :1/2),\ -0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148936 + 0.767296i\)
\(L(\frac12)\) \(\approx\) \(0.148936 + 0.767296i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-3.82 - 1.54i)T \)
43 \( 1 + T \)
good2 \( 1 + 1.28T + 2T^{2} \)
3 \( 1 - 2.62iT - 3T^{2} \)
5 \( 1 + 1.51iT - 5T^{2} \)
7 \( 1 - 3.46iT - 7T^{2} \)
11 \( 1 - 2.66iT - 11T^{2} \)
13 \( 1 - 6.10T + 13T^{2} \)
19 \( 1 + 5.46T + 19T^{2} \)
23 \( 1 - 4.54iT - 23T^{2} \)
29 \( 1 - 5.97iT - 29T^{2} \)
31 \( 1 + 4.36iT - 31T^{2} \)
37 \( 1 + 7.49iT - 37T^{2} \)
41 \( 1 - 2.38iT - 41T^{2} \)
47 \( 1 - 4.98T + 47T^{2} \)
53 \( 1 + 13.2T + 53T^{2} \)
59 \( 1 + 15.3T + 59T^{2} \)
61 \( 1 + 3.31iT - 61T^{2} \)
67 \( 1 + 2.45T + 67T^{2} \)
71 \( 1 + 12.5iT - 71T^{2} \)
73 \( 1 - 1.13iT - 73T^{2} \)
79 \( 1 - 6.56iT - 79T^{2} \)
83 \( 1 - 9.61T + 83T^{2} \)
89 \( 1 + 14.4T + 89T^{2} \)
97 \( 1 - 16.0iT - 97T^{2} \)
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

−10.63862276116467436591856769124, −9.605763613622289770625962437035, −9.100006276487028593335247777453, −8.677030001026837169097845811910, −7.82519321112420570303764141998, −6.10532500502435365912804348313, −5.22524011670266613346380968423, −4.43374951063345509077736610194, −3.47691699595932209264921901999, −1.64301566180824059225686871564, 0.63118964505756506355144182570, 1.47796196792248233313411038365, 3.14705605632133788438810859656, 4.35331340965573950594765346319, 6.12832826833936579687398164455, 6.68154468243735054266076460383, 7.55432927768462522635790304891, 8.205405439183980792417937334395, 8.804613568014963732751805693797, 10.23387055384253382479781936138

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