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Properties

Label 2-731-17.16-c1-0-44
Degree $2$
Conductor $731$
Sign $0.487 - 0.873i$
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  + 2.64·2-s + 1.50i·3-s + 4.99·4-s + 2.09i·5-s + 3.98i·6-s + 1.03i·7-s + 7.91·8-s + 0.733·9-s + 5.53i·10-s − 5.77i·11-s + 7.51i·12-s − 7.17·13-s + 2.74i·14-s − 3.15·15-s + 10.9·16-s + (2.00 − 3.60i)17-s + ⋯
L(s)  = 1  + 1.86·2-s + 0.869i·3-s + 2.49·4-s + 0.936i·5-s + 1.62i·6-s + 0.391i·7-s + 2.79·8-s + 0.244·9-s + 1.75i·10-s − 1.74i·11-s + 2.17i·12-s − 1.99·13-s + 0.732i·14-s − 0.814·15-s + 2.73·16-s + (0.487 − 0.873i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.487 - 0.873i$
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: \((2,\ 731,\ (\ :1/2),\ 0.487 - 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.91862 + 2.30032i\)
\(L(\frac12)\) \(\approx\) \(3.91862 + 2.30032i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-2.00 + 3.60i)T \)
43 \( 1 - T \)
good2 \( 1 - 2.64T + 2T^{2} \)
3 \( 1 - 1.50iT - 3T^{2} \)
5 \( 1 - 2.09iT - 5T^{2} \)
7 \( 1 - 1.03iT - 7T^{2} \)
11 \( 1 + 5.77iT - 11T^{2} \)
13 \( 1 + 7.17T + 13T^{2} \)
19 \( 1 + 3.98T + 19T^{2} \)
23 \( 1 + 2.14iT - 23T^{2} \)
29 \( 1 - 2.72iT - 29T^{2} \)
31 \( 1 - 1.82iT - 31T^{2} \)
37 \( 1 - 0.454iT - 37T^{2} \)
41 \( 1 + 5.95iT - 41T^{2} \)
47 \( 1 - 1.22T + 47T^{2} \)
53 \( 1 - 1.68T + 53T^{2} \)
59 \( 1 - 7.11T + 59T^{2} \)
61 \( 1 - 10.4iT - 61T^{2} \)
67 \( 1 + 9.81T + 67T^{2} \)
71 \( 1 + 7.76iT - 71T^{2} \)
73 \( 1 + 0.449iT - 73T^{2} \)
79 \( 1 - 6.17iT - 79T^{2} \)
83 \( 1 + 9.01T + 83T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 + 18.2iT - 97T^{2} \)
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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

−10.65435235519737002512136586188, −10.22109775195546761760034220463, −8.903978620745770014338476270819, −7.40894508404413100771665672311, −6.80129120653031035424038594957, −5.72040270765719131506248805647, −5.05767438156185516816735743443, −4.12706909406761870300854857671, −3.10473874057258965591083935711, −2.54043273787650982241439311106, 1.66451777588715081052109485296, 2.45499961725328245924248250796, 4.23020372444632620338791932247, 4.58399374502450928341135712405, 5.53303497626750046894006759997, 6.72078451890553396468397322539, 7.27379433004375088808806858707, 7.947468318390038136233195021675, 9.689360748852682730977534434795, 10.36400526612213898741465436658

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