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Properties

Label 2-731-731.558-c1-0-59
Degree $2$
Conductor $731$
Sign $-0.711 + 0.703i$
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 + (−4.38 + 0.872i)11-s + (−4.77 − 4.77i)13-s − 4.00i·16-s + (−2.07 − 3.56i)17-s + (1.52 + 7.64i)23-s + (1.91 − 4.61i)25-s + (10.2 + 2.03i)31-s + (−5.54 + 2.29i)36-s + (−10.0 − 6.70i)41-s + (−6.05 − 2.50i)43-s + (−4.97 + 7.44i)44-s + (6.16 + 6.16i)47-s + (2.67 + 6.46i)49-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)4-s + (−0.923 − 0.382i)9-s + (−1.32 + 0.263i)11-s + (−1.32 − 1.32i)13-s − 1.00i·16-s + (−0.502 − 0.864i)17-s + (0.317 + 1.59i)23-s + (0.382 − 0.923i)25-s + (1.84 + 0.366i)31-s + (−0.923 + 0.382i)36-s + (−1.56 − 1.04i)41-s + (−0.923 − 0.382i)43-s + (−0.749 + 1.12i)44-s + (0.898 + 0.898i)47-s + (0.382 + 0.923i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.711 + 0.703i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (558, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.711 + 0.703i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.367470 - 0.894426i\)
\(L(\frac12)\) \(\approx\) \(0.367470 - 0.894426i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (2.07 + 3.56i)T \)
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 \( 1 + (4.38 - 0.872i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (4.77 + 4.77i)T + 13iT^{2} \)
19 \( 1 + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-1.52 - 7.64i)T + (-21.2 + 8.80i)T^{2} \)
29 \( 1 + (11.0 - 26.7i)T^{2} \)
31 \( 1 + (-10.2 - 2.03i)T + (28.6 + 11.8i)T^{2} \)
37 \( 1 + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (10.0 + 6.70i)T + (15.6 + 37.8i)T^{2} \)
47 \( 1 + (-6.16 - 6.16i)T + 47iT^{2} \)
53 \( 1 + (-10.1 + 4.22i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-3.46 + 8.36i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (23.3 + 56.3i)T^{2} \)
67 \( 1 + 11.3iT - 67T^{2} \)
71 \( 1 + (65.5 + 27.1i)T^{2} \)
73 \( 1 + (-27.9 + 67.4i)T^{2} \)
79 \( 1 + (-9.03 + 1.79i)T + (72.9 - 30.2i)T^{2} \)
83 \( 1 + (2.50 + 6.05i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + 89iT^{2} \)
97 \( 1 + (-1.58 - 2.37i)T + (-37.1 + 89.6i)T^{2} \)
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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

−10.21016670825225218775024689672, −9.427578834247746463050709065274, −8.198706890359962021276411764356, −7.44295563965590973947249908106, −6.55093474340671184324090532934, −5.32937969484098975246824400751, −5.09957847238062074751101362596, −3.01721708468101531013998040064, −2.43579154902726021476492979937, −0.44033068918331144713624285261, 2.27237243837536742143792983319, 2.83027011566362021345788607150, 4.32108625254821952779104924156, 5.30108663664572023393021741098, 6.51643652807600702783285577667, 7.17532130982921236183237540132, 8.303506466371751615017087613733, 8.608800791518867945384301759302, 10.07269735903560581797440595277, 10.73580007837636831079779815375

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