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Properties

Label 2-731-17.2-c1-0-31
Degree $2$
Conductor $731$
Sign $-0.0516 + 0.998i$
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  + (−1.15 − 1.15i)2-s + (0.645 − 0.267i)3-s + 0.662i·4-s + (0.125 + 0.302i)5-s + (−1.05 − 0.436i)6-s + (0.205 − 0.496i)7-s + (−1.54 + 1.54i)8-s + (−1.77 + 1.77i)9-s + (0.204 − 0.493i)10-s + (5.51 + 2.28i)11-s + (0.177 + 0.427i)12-s − 3.02i·13-s + (−0.810 + 0.335i)14-s + (0.161 + 0.161i)15-s + 4.88·16-s + (0.0210 − 4.12i)17-s + ⋯
L(s)  = 1  + (−0.815 − 0.815i)2-s + (0.372 − 0.154i)3-s + 0.331i·4-s + (0.0560 + 0.135i)5-s + (−0.429 − 0.178i)6-s + (0.0777 − 0.187i)7-s + (−0.545 + 0.545i)8-s + (−0.592 + 0.592i)9-s + (0.0646 − 0.156i)10-s + (1.66 + 0.689i)11-s + (0.0511 + 0.123i)12-s − 0.839i·13-s + (−0.216 + 0.0897i)14-s + (0.0417 + 0.0417i)15-s + 1.22·16-s + (0.00511 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0516 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0516 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.0516 + 0.998i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.0516 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.780989 - 0.822398i\)
\(L(\frac12)\) \(\approx\) \(0.780989 - 0.822398i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-0.0210 + 4.12i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (1.15 + 1.15i)T + 2iT^{2} \)
3 \( 1 + (-0.645 + 0.267i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (-0.125 - 0.302i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-0.205 + 0.496i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-5.51 - 2.28i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 3.02iT - 13T^{2} \)
19 \( 1 + (1.54 + 1.54i)T + 19iT^{2} \)
23 \( 1 + (-6.80 - 2.82i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (3.74 + 9.05i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-7.55 + 3.13i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (7.45 - 3.08i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-2.26 + 5.47i)T + (-28.9 - 28.9i)T^{2} \)
47 \( 1 - 5.04iT - 47T^{2} \)
53 \( 1 + (-4.11 - 4.11i)T + 53iT^{2} \)
59 \( 1 + (-8.90 + 8.90i)T - 59iT^{2} \)
61 \( 1 + (1.54 - 3.72i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 9.13T + 67T^{2} \)
71 \( 1 + (-7.17 + 2.97i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (2.55 + 6.16i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (10.1 + 4.21i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-3.94 - 3.94i)T + 83iT^{2} \)
89 \( 1 - 4.71iT - 89T^{2} \)
97 \( 1 + (1.78 + 4.31i)T + (-68.5 + 68.5i)T^{2} \)
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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

−10.10844308815702440235178154057, −9.319004701595498063635504673429, −8.766303404125970326438379000738, −7.80913156635344582998360304583, −6.86097925018023897586569571990, −5.71885137122146915254160649814, −4.55556137298993843831463606681, −3.10815291649101454793457340788, −2.23747714628164040588445318429, −0.897280664815903877261548623292, 1.24355657151790729779744836460, 3.20716183833941176666261564662, 3.98744801558331737639512620747, 5.56286439486823086762278955854, 6.62987963297043696907102180887, 6.91159606518901977417725901712, 8.490177053329247158613627800908, 8.797815784080943866830879120169, 9.172361512779716154718417251349, 10.35291672456516268164075591073

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