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Properties

Label 2-731-17.16-c1-0-35
Degree $2$
Conductor $731$
Sign $0.940 - 0.340i$
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  + 2.20·2-s − 1.64i·3-s + 2.87·4-s + 3.04i·5-s − 3.62i·6-s + 3.21i·7-s + 1.92·8-s + 0.299·9-s + 6.72i·10-s + 1.91i·11-s − 4.72i·12-s + 2.30·13-s + 7.10i·14-s + 5.00·15-s − 1.48·16-s + (3.87 − 1.40i)17-s + ⋯
L(s)  = 1  + 1.56·2-s − 0.948i·3-s + 1.43·4-s + 1.36i·5-s − 1.48i·6-s + 1.21i·7-s + 0.682·8-s + 0.0996·9-s + 2.12i·10-s + 0.578i·11-s − 1.36i·12-s + 0.637·13-s + 1.89i·14-s + 1.29·15-s − 0.372·16-s + (0.940 − 0.340i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.340i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.940 - 0.340i$
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: \((2,\ 731,\ (\ :1/2),\ 0.940 - 0.340i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.53439 + 0.619615i\)
\(L(\frac12)\) \(\approx\) \(3.53439 + 0.619615i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-3.87 + 1.40i)T \)
43 \( 1 - T \)
good2 \( 1 - 2.20T + 2T^{2} \)
3 \( 1 + 1.64iT - 3T^{2} \)
5 \( 1 - 3.04iT - 5T^{2} \)
7 \( 1 - 3.21iT - 7T^{2} \)
11 \( 1 - 1.91iT - 11T^{2} \)
13 \( 1 - 2.30T + 13T^{2} \)
19 \( 1 + 3.35T + 19T^{2} \)
23 \( 1 + 5.90iT - 23T^{2} \)
29 \( 1 + 8.26iT - 29T^{2} \)
31 \( 1 - 4.33iT - 31T^{2} \)
37 \( 1 + 1.16iT - 37T^{2} \)
41 \( 1 + 0.352iT - 41T^{2} \)
47 \( 1 + 8.52T + 47T^{2} \)
53 \( 1 + 2.93T + 53T^{2} \)
59 \( 1 - 8.17T + 59T^{2} \)
61 \( 1 + 1.93iT - 61T^{2} \)
67 \( 1 - 5.26T + 67T^{2} \)
71 \( 1 - 11.3iT - 71T^{2} \)
73 \( 1 + 11.3iT - 73T^{2} \)
79 \( 1 - 5.46iT - 79T^{2} \)
83 \( 1 - 16.8T + 83T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 + 5.89iT - 97T^{2} \)
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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

−10.78734036407877239590286724382, −9.794818913196841505901272524012, −8.441631798128117369033022234175, −7.42222511393599064726813709147, −6.42377995587920647220622909168, −6.29357170106344199330037693645, −5.09353972300184826084323365791, −3.87812658194831297117709270298, −2.74014294150829870228843994957, −2.13455540234916491768372965464, 1.32208154628926922499761208686, 3.54591596310375956427803199914, 3.86719519333401282565911504263, 4.82843925441288229848324420526, 5.37110356147046772801650333205, 6.43828917544321150101735012647, 7.63562660818198882634177206053, 8.713718841500578785738738020208, 9.587760631782529121477129853321, 10.56418014247212716464979634578

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