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Properties

Label 2-731-731.135-c1-0-5
Degree $2$
Conductor $731$
Sign $-0.663 - 0.747i$
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  − 2.09·2-s + (−0.414 − 0.239i)3-s + 2.37·4-s + (1.22 + 0.706i)5-s + (0.866 + 0.500i)6-s + (−2.02 + 1.16i)7-s − 0.786·8-s + (−1.38 − 2.39i)9-s + (−2.55 − 1.47i)10-s − 2.22i·11-s + (−0.984 − 0.568i)12-s + (3.44 + 5.96i)13-s + (4.23 − 2.44i)14-s + (−0.337 − 0.585i)15-s − 3.10·16-s + (−3.88 + 1.38i)17-s + ⋯
L(s)  = 1  − 1.47·2-s + (−0.239 − 0.138i)3-s + 1.18·4-s + (0.546 + 0.315i)5-s + (0.353 + 0.204i)6-s + (−0.765 + 0.441i)7-s − 0.278·8-s + (−0.461 − 0.799i)9-s + (−0.809 − 0.467i)10-s − 0.669i·11-s + (−0.284 − 0.164i)12-s + (0.954 + 1.65i)13-s + (1.13 − 0.653i)14-s + (−0.0872 − 0.151i)15-s − 0.776·16-s + (−0.941 + 0.336i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 - 0.747i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.663 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.663 - 0.747i$
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: \((2,\ 731,\ (\ :1/2),\ -0.663 - 0.747i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.114663 + 0.255081i\)
\(L(\frac12)\) \(\approx\) \(0.114663 + 0.255081i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (3.88 - 1.38i)T \)
43 \( 1 + (-5.40 + 3.71i)T \)
good2 \( 1 + 2.09T + 2T^{2} \)
3 \( 1 + (0.414 + 0.239i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.22 - 0.706i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.02 - 1.16i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 2.22iT - 11T^{2} \)
13 \( 1 + (-3.44 - 5.96i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + (-0.0342 + 0.0593i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.19 - 1.84i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.97 + 1.14i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.10 + 2.36i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.36 + 3.67i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.1iT - 41T^{2} \)
47 \( 1 + 0.393T + 47T^{2} \)
53 \( 1 + (3.16 - 5.47i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.31T + 59T^{2} \)
61 \( 1 + (11.8 - 6.81i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.37 - 7.58i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.52 - 4.92i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.29 - 4.21i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (10.7 - 6.22i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.32 + 10.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.28 - 3.95i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.0iT - 97T^{2} \)
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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

−10.60188419758886761732762298638, −9.539720726718488084932523702029, −9.016220802624393177308836805400, −8.606419995359242328773225236247, −7.19056151835975128526374497111, −6.38744028675388108366254240341, −5.98129279729986497840200075860, −4.12157831211568879639634922023, −2.73570207451005782486695352509, −1.44254216316802151989871450329, 0.24797084560029567470473679334, 1.75635815891817873722376776986, 3.12799580645354670614738378485, 4.76319847061196558653281648664, 5.75560995968192465270808599180, 6.84143672759217750340330985382, 7.64535636017219065763421746911, 8.585838407366768886342396352072, 9.170506551316673811186204329936, 10.13941922918346567720860741110

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