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Properties

Label 2-731-43.6-c1-0-13
Degree $2$
Conductor $731$
Sign $-0.691 - 0.722i$
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  − 0.966·2-s + (−0.835 + 1.44i)3-s − 1.06·4-s + (−0.811 + 1.40i)5-s + (0.806 − 1.39i)6-s + (−1.19 − 2.07i)7-s + 2.96·8-s + (0.104 + 0.181i)9-s + (0.783 − 1.35i)10-s + 5.63·11-s + (0.891 − 1.54i)12-s + (1.57 + 2.73i)13-s + (1.15 + 2.00i)14-s + (−1.35 − 2.34i)15-s − 0.728·16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  − 0.683·2-s + (−0.482 + 0.835i)3-s − 0.533·4-s + (−0.362 + 0.628i)5-s + (0.329 − 0.570i)6-s + (−0.452 − 0.784i)7-s + 1.04·8-s + (0.0349 + 0.0605i)9-s + (0.247 − 0.429i)10-s + 1.69·11-s + (0.257 − 0.445i)12-s + (0.438 + 0.758i)13-s + (0.309 + 0.535i)14-s + (−0.350 − 0.606i)15-s − 0.182·16-s + (0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.691 - 0.722i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.691 - 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.244183 + 0.572192i\)
\(L(\frac12)\) \(\approx\) \(0.244183 + 0.572192i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (2.90 + 5.87i)T \)
good2 \( 1 + 0.966T + 2T^{2} \)
3 \( 1 + (0.835 - 1.44i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.811 - 1.40i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.19 + 2.07i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 5.63T + 11T^{2} \)
13 \( 1 + (-1.57 - 2.73i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + (0.504 - 0.873i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.36 + 2.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.40 - 2.43i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.44 - 2.49i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.99 - 3.44i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.86T + 41T^{2} \)
47 \( 1 + 1.65T + 47T^{2} \)
53 \( 1 + (-0.0801 + 0.138i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 4.21T + 59T^{2} \)
61 \( 1 + (-0.614 - 1.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.94 - 3.37i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.92 - 10.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.47 - 6.02i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.80 - 8.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.76 + 3.04i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (9.13 - 15.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.05T + 97T^{2} \)
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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

−10.59204958406583992986279396841, −9.877867703297091538041564370033, −9.180853262313457880180088738656, −8.370949222932672383274639268283, −7.09434630290239688611044742274, −6.62367261872769044225985690908, −5.13711376640550432547329063545, −4.04911086296702855101793381828, −3.70430977665486793010955546481, −1.34781610433145824515432391129, 0.53130312947793671822146309410, 1.54645631461296491186113329453, 3.52630903857067012044223733778, 4.61328834346248937397028124991, 5.81106565026288837371025931397, 6.57994168353015558540162692750, 7.59691354458918633818727301523, 8.482378064205115732808640755949, 9.180006921877571653529237614767, 9.707524329159525420972391839172

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