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Properties

Label 2-731-731.601-c1-0-36
Degree $2$
Conductor $731$
Sign $0.998 + 0.0522i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 + 1.41i)4-s + (2.77 − 1.14i)9-s + (1.17 − 5.89i)11-s + (1.77 − 1.77i)13-s + 4.00i·16-s + (−3.98 − 1.05i)17-s + (7.98 + 1.58i)23-s + (−1.91 − 4.61i)25-s + (2.17 + 10.9i)31-s + (5.54 + 2.29i)36-s + (4.76 + 7.13i)41-s + (−6.05 + 2.50i)43-s + (9.99 − 6.67i)44-s + (0.935 − 0.935i)47-s + (−2.67 + 6.46i)49-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)4-s + (0.923 − 0.382i)9-s + (0.353 − 1.77i)11-s + (0.493 − 0.493i)13-s + 1.00i·16-s + (−0.966 − 0.255i)17-s + (1.66 + 0.331i)23-s + (−0.382 − 0.923i)25-s + (0.389 + 1.95i)31-s + (0.923 + 0.382i)36-s + (0.744 + 1.11i)41-s + (−0.923 + 0.382i)43-s + (1.50 − 1.00i)44-s + (0.136 − 0.136i)47-s + (−0.382 + 0.923i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.998 + 0.0522i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.998 + 0.0522i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95880 - 0.0512224i\)
\(L(\frac12)\) \(\approx\) \(1.95880 - 0.0512224i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (3.98 + 1.05i)T \)
43 \( 1 + (6.05 - 2.50i)T \)
good2 \( 1 + (-1.41 - 1.41i)T^{2} \)
3 \( 1 + (-2.77 + 1.14i)T^{2} \)
5 \( 1 + (1.91 + 4.61i)T^{2} \)
7 \( 1 + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (-1.17 + 5.89i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + (-1.77 + 1.77i)T - 13iT^{2} \)
19 \( 1 + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-7.98 - 1.58i)T + (21.2 + 8.80i)T^{2} \)
29 \( 1 + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (-2.17 - 10.9i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-4.76 - 7.13i)T + (-15.6 + 37.8i)T^{2} \)
47 \( 1 + (-0.935 + 0.935i)T - 47iT^{2} \)
53 \( 1 + (1.00 + 0.414i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (5.80 + 14.0i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + 16.3iT - 67T^{2} \)
71 \( 1 + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (2.05 - 10.3i)T + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (2.50 - 6.05i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 - 89iT^{2} \)
97 \( 1 + (-15.8 - 10.6i)T + (37.1 + 89.6i)T^{2} \)
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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

−10.77048161184473589099307608225, −9.387494257156118219730139786179, −8.570058832255667602169144734663, −7.87213751751247918140183120244, −6.65780239566044025192647334746, −6.35319261584338411906399166128, −4.87470092691964118422788739716, −3.60420476266569301259935374081, −2.93660396385215334960513066360, −1.22028323507647595613900731734, 1.50637238504644656825039950127, 2.33414469757957925729261403225, 4.13362769801832389696259986490, 4.87143150160277181182949407965, 6.08289668576294805050362058196, 7.13786854223794080689356397927, 7.28597924302888230546883221320, 8.907333745510485612077318532157, 9.679047284035662707445223453825, 10.35274645361097507612404909008

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