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Properties

Label 2-731-17.16-c1-0-0
Degree $2$
Conductor $731$
Sign $-0.973 - 0.229i$
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  − 1.26·2-s − 2.08i·3-s − 0.390·4-s + 3.03i·5-s + 2.64i·6-s + 3.31i·7-s + 3.03·8-s − 1.34·9-s − 3.84i·10-s − 0.560i·11-s + 0.813i·12-s − 2.04·13-s − 4.20i·14-s + 6.31·15-s − 3.06·16-s + (−4.01 − 0.945i)17-s + ⋯
L(s)  = 1  − 0.897·2-s − 1.20i·3-s − 0.195·4-s + 1.35i·5-s + 1.07i·6-s + 1.25i·7-s + 1.07·8-s − 0.449·9-s − 1.21i·10-s − 0.169i·11-s + 0.234i·12-s − 0.566·13-s − 1.12i·14-s + 1.63·15-s − 0.766·16-s + (−0.973 − 0.229i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.973 - 0.229i$
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: \((2,\ 731,\ (\ :1/2),\ -0.973 - 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00898454 + 0.0773037i\)
\(L(\frac12)\) \(\approx\) \(0.00898454 + 0.0773037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (4.01 + 0.945i)T \)
43 \( 1 - T \)
good2 \( 1 + 1.26T + 2T^{2} \)
3 \( 1 + 2.08iT - 3T^{2} \)
5 \( 1 - 3.03iT - 5T^{2} \)
7 \( 1 - 3.31iT - 7T^{2} \)
11 \( 1 + 0.560iT - 11T^{2} \)
13 \( 1 + 2.04T + 13T^{2} \)
19 \( 1 + 6.66T + 19T^{2} \)
23 \( 1 + 1.92iT - 23T^{2} \)
29 \( 1 - 2.52iT - 29T^{2} \)
31 \( 1 + 5.60iT - 31T^{2} \)
37 \( 1 + 3.90iT - 37T^{2} \)
41 \( 1 + 3.47iT - 41T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 - 4.62T + 53T^{2} \)
59 \( 1 + 13.8T + 59T^{2} \)
61 \( 1 - 12.4iT - 61T^{2} \)
67 \( 1 - 2.81T + 67T^{2} \)
71 \( 1 - 4.34iT - 71T^{2} \)
73 \( 1 - 14.3iT - 73T^{2} \)
79 \( 1 + 11.3iT - 79T^{2} \)
83 \( 1 + 7.69T + 83T^{2} \)
89 \( 1 + 2.96T + 89T^{2} \)
97 \( 1 - 5.44iT - 97T^{2} \)
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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

−10.75195572560577231082475413584, −9.870352962803869676140098241644, −8.883463003549948047527511720533, −8.269615103732901203491872381618, −7.29746554402829045816786657259, −6.71913228794937920809026375244, −5.85855266595472360340495670086, −4.37580775432925708777386145231, −2.62439971682036340108301859422, −1.98055484271708353867682757578, 0.05458900431066480139266296787, 1.55283816146413691244469696927, 3.82119420454010754947815314224, 4.66999182950625387259953598222, 4.82295842819768484369698281993, 6.63041050367197946619141764676, 7.78316568264949495640773764695, 8.553556141251502643174941452779, 9.223707002124740607347737044781, 9.888309219172990045144939279122

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