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Properties

Label 2-731-731.208-c1-0-46
Degree $2$
Conductor $731$
Sign $0.0293 + 0.999i$
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  − 1.62i·2-s + (0.474 + 0.127i)3-s − 0.651·4-s + (2.37 + 0.637i)5-s + (0.207 − 0.773i)6-s + (4.72 − 1.26i)7-s − 2.19i·8-s + (−2.38 − 1.37i)9-s + (1.03 − 3.87i)10-s + (0.797 + 0.797i)11-s + (−0.309 − 0.0829i)12-s + (−2.17 + 3.76i)13-s + (−2.05 − 7.68i)14-s + (1.04 + 0.605i)15-s − 4.87·16-s + (−2.22 + 3.47i)17-s + ⋯
L(s)  = 1  − 1.15i·2-s + (0.274 + 0.0734i)3-s − 0.325·4-s + (1.06 + 0.285i)5-s + (0.0846 − 0.315i)6-s + (1.78 − 0.478i)7-s − 0.776i·8-s + (−0.796 − 0.459i)9-s + (0.328 − 1.22i)10-s + (0.240 + 0.240i)11-s + (−0.0893 − 0.0239i)12-s + (−0.602 + 1.04i)13-s + (−0.550 − 2.05i)14-s + (0.270 + 0.156i)15-s − 1.21·16-s + (−0.538 + 0.842i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0293 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0293 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.0293 + 0.999i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.0293 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70501 - 1.65563i\)
\(L(\frac12)\) \(\approx\) \(1.70501 - 1.65563i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (2.22 - 3.47i)T \)
43 \( 1 + (6.29 + 1.83i)T \)
good2 \( 1 + 1.62iT - 2T^{2} \)
3 \( 1 + (-0.474 - 0.127i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-2.37 - 0.637i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-4.72 + 1.26i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.797 - 0.797i)T + 11iT^{2} \)
13 \( 1 + (2.17 - 3.76i)T + (-6.5 - 11.2i)T^{2} \)
19 \( 1 + (-7.38 + 4.26i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.586 - 2.19i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.33 - 8.70i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (7.40 + 1.98i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (4.59 + 1.23i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.206 - 0.206i)T + 41iT^{2} \)
47 \( 1 - 11.0T + 47T^{2} \)
53 \( 1 + (5.56 - 3.21i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 4.07iT - 59T^{2} \)
61 \( 1 + (10.9 - 2.92i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (4.01 + 6.94i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.793 + 2.96i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.341 + 1.27i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.42 - 1.45i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-7.37 + 4.25i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.70 - 2.96i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.41 - 5.41i)T - 97iT^{2} \)
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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

−10.42801325166872261809400141625, −9.300375014013787244532390954195, −8.994017436420295349251295363108, −7.54557219372091652964612709662, −6.77246906296414053398548597217, −5.47419447116662806241561010189, −4.51043915745489309060669722813, −3.34457279050024205771506812703, −2.11568730136012148386235097324, −1.49404166670406232923991231322, 1.77351488992058496626483131035, 2.71267703498524312568578052036, 4.84100530370090938803289885404, 5.47597342993097646130915801900, 5.84047010127344193416469594626, 7.36893597519319614901233726985, 7.931855058598277430139249177709, 8.592039483652620985216293412825, 9.412757973016800051213852534168, 10.57530641998986497227224630001

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