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Properties

Label 2-731-731.135-c1-0-44
Degree $2$
Conductor $731$
Sign $-0.615 + 0.787i$
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  − 1.73·2-s + (1.10 + 0.638i)3-s + 1.02·4-s + (−2.28 − 1.32i)5-s + (−1.92 − 1.11i)6-s + (4.26 − 2.46i)7-s + 1.69·8-s + (−0.683 − 1.18i)9-s + (3.98 + 2.29i)10-s − 0.464i·11-s + (1.13 + 0.654i)12-s + (−2.87 − 4.97i)13-s + (−7.41 + 4.27i)14-s + (−1.68 − 2.92i)15-s − 4.99·16-s + (−0.621 + 4.07i)17-s + ⋯
L(s)  = 1  − 1.22·2-s + (0.638 + 0.368i)3-s + 0.512·4-s + (−1.02 − 0.590i)5-s + (−0.785 − 0.453i)6-s + (1.61 − 0.929i)7-s + 0.599·8-s + (−0.227 − 0.394i)9-s + (1.25 + 0.726i)10-s − 0.140i·11-s + (0.327 + 0.189i)12-s + (−0.797 − 1.38i)13-s + (−1.98 + 1.14i)14-s + (−0.435 − 0.754i)15-s − 1.24·16-s + (−0.150 + 0.988i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.615 + 0.787i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.615 + 0.787i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.245097 - 0.502738i\)
\(L(\frac12)\) \(\approx\) \(0.245097 - 0.502738i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (0.621 - 4.07i)T \)
43 \( 1 + (-1.89 + 6.27i)T \)
good2 \( 1 + 1.73T + 2T^{2} \)
3 \( 1 + (-1.10 - 0.638i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.28 + 1.32i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-4.26 + 2.46i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 0.464iT - 11T^{2} \)
13 \( 1 + (2.87 + 4.97i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + (2.00 - 3.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.36 - 1.36i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.94 - 4.01i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.73 + 3.31i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.115 - 0.0666i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.696iT - 41T^{2} \)
47 \( 1 + 3.34T + 47T^{2} \)
53 \( 1 + (-0.460 + 0.797i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.34T + 59T^{2} \)
61 \( 1 + (1.33 - 0.768i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.99 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (10.6 - 6.12i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.24 + 4.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (14.4 - 8.32i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.45 + 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.204 - 0.353i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.60iT - 97T^{2} \)
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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

−10.04118913813148729937918279678, −8.969353308547943806432969901625, −8.316775148813939813281977836738, −7.87277003789502472114901683427, −7.32100415314571632172106979342, −5.40874545626990874096746660461, −4.35469534236785755384036070322, −3.65988462543044616664145716866, −1.74129640850136444211692473299, −0.40594058767603065321945381831, 1.81819909867973941445215060104, 2.55692912659850445366290678274, 4.36018155966848420393554112139, 5.12039447817519391376884787786, 7.05455863478909343975878866549, 7.46536507284924782664414979212, 8.165562682155407027448352381163, 8.914482321733648431243472617236, 9.391218901382170008305777130119, 10.89518792543958922722054097549

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