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Properties

Label 2-731-17.16-c1-0-22
Degree $2$
Conductor $731$
Sign $0.0672 - 0.997i$
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  − 1.90·2-s + 2.97i·3-s + 1.61·4-s + 1.24i·5-s − 5.65i·6-s − 4.11i·7-s + 0.734·8-s − 5.85·9-s − 2.36i·10-s − 1.10i·11-s + 4.80i·12-s + 5.49·13-s + 7.81i·14-s − 3.70·15-s − 4.62·16-s + (−0.277 + 4.11i)17-s + ⋯
L(s)  = 1  − 1.34·2-s + 1.71i·3-s + 0.806·4-s + 0.557i·5-s − 2.30i·6-s − 1.55i·7-s + 0.259·8-s − 1.95·9-s − 0.749i·10-s − 0.334i·11-s + 1.38i·12-s + 1.52·13-s + 2.08i·14-s − 0.957·15-s − 1.15·16-s + (−0.0672 + 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0672 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0672 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.0672 - 0.997i$
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: \((2,\ 731,\ (\ :1/2),\ 0.0672 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.576685 + 0.539130i\)
\(L(\frac12)\) \(\approx\) \(0.576685 + 0.539130i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (0.277 - 4.11i)T \)
43 \( 1 + T \)
good2 \( 1 + 1.90T + 2T^{2} \)
3 \( 1 - 2.97iT - 3T^{2} \)
5 \( 1 - 1.24iT - 5T^{2} \)
7 \( 1 + 4.11iT - 7T^{2} \)
11 \( 1 + 1.10iT - 11T^{2} \)
13 \( 1 - 5.49T + 13T^{2} \)
19 \( 1 - 8.40T + 19T^{2} \)
23 \( 1 + 3.45iT - 23T^{2} \)
29 \( 1 - 1.83iT - 29T^{2} \)
31 \( 1 - 3.35iT - 31T^{2} \)
37 \( 1 + 3.35iT - 37T^{2} \)
41 \( 1 - 0.843iT - 41T^{2} \)
47 \( 1 + 7.88T + 47T^{2} \)
53 \( 1 - 6.03T + 53T^{2} \)
59 \( 1 - 8.31T + 59T^{2} \)
61 \( 1 + 8.77iT - 61T^{2} \)
67 \( 1 + 6.67T + 67T^{2} \)
71 \( 1 - 6.75iT - 71T^{2} \)
73 \( 1 - 9.72iT - 73T^{2} \)
79 \( 1 + 10.5iT - 79T^{2} \)
83 \( 1 - 3.49T + 83T^{2} \)
89 \( 1 - 2.18T + 89T^{2} \)
97 \( 1 - 2.51iT - 97T^{2} \)
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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

−10.32687988514336401998717660234, −10.01179969811794618072111511103, −8.975614006156202322989638778421, −8.380099751367504233731828517457, −7.37610825714396417113553207325, −6.37655280617672430332614379334, −5.01480524073630393119638461850, −3.92121780389234511787959465141, −3.32397897981319861507479144590, −1.04167752437176776442936942773, 0.906496618550791263694634523091, 1.76296034528437889415331270387, 2.93088794223759773131317646804, 5.17228911060311512189358902064, 6.00773330976996091942881899126, 7.02267129390787413078449336949, 7.76864539638971015423755600788, 8.522262517691896278663080700902, 9.015338003914231894821914631910, 9.772613145849377560803732885736

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