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Properties

Label 2-731-17.2-c1-0-55
Degree $2$
Conductor $731$
Sign $0.0661 + 0.997i$
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  + (0.493 + 0.493i)2-s + (2.20 − 0.912i)3-s − 1.51i·4-s + (−1.26 − 3.04i)5-s + (1.53 + 0.636i)6-s + (−1.60 + 3.86i)7-s + (1.73 − 1.73i)8-s + (1.89 − 1.89i)9-s + (0.879 − 2.12i)10-s + (0.798 + 0.330i)11-s + (−1.38 − 3.33i)12-s − 4.97i·13-s + (−2.69 + 1.11i)14-s + (−5.55 − 5.55i)15-s − 1.31·16-s + (−0.463 − 4.09i)17-s + ⋯
L(s)  = 1  + (0.348 + 0.348i)2-s + (1.27 − 0.526i)3-s − 0.756i·4-s + (−0.563 − 1.36i)5-s + (0.627 + 0.259i)6-s + (−0.605 + 1.46i)7-s + (0.612 − 0.612i)8-s + (0.632 − 0.632i)9-s + (0.278 − 0.671i)10-s + (0.240 + 0.0997i)11-s + (−0.398 − 0.962i)12-s − 1.37i·13-s + (−0.721 + 0.298i)14-s + (−1.43 − 1.43i)15-s − 0.328·16-s + (−0.112 − 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0661 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0661 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.0661 + 0.997i$
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: \((2,\ 731,\ (\ :1/2),\ 0.0661 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70858 - 1.59900i\)
\(L(\frac12)\) \(\approx\) \(1.70858 - 1.59900i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (0.463 + 4.09i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (-0.493 - 0.493i)T + 2iT^{2} \)
3 \( 1 + (-2.20 + 0.912i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (1.26 + 3.04i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (1.60 - 3.86i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.798 - 0.330i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 4.97iT - 13T^{2} \)
19 \( 1 + (-0.523 - 0.523i)T + 19iT^{2} \)
23 \( 1 + (-6.47 - 2.68i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (0.419 + 1.01i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (1.54 - 0.638i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-1.02 + 0.422i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.36 - 3.30i)T + (-28.9 - 28.9i)T^{2} \)
47 \( 1 - 4.76iT - 47T^{2} \)
53 \( 1 + (-0.708 - 0.708i)T + 53iT^{2} \)
59 \( 1 + (-5.32 + 5.32i)T - 59iT^{2} \)
61 \( 1 + (-0.461 + 1.11i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 - 14.3T + 67T^{2} \)
71 \( 1 + (-9.57 + 3.96i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-3.54 - 8.54i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-5.43 - 2.25i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-7.05 - 7.05i)T + 83iT^{2} \)
89 \( 1 + 6.34iT - 89T^{2} \)
97 \( 1 + (0.907 + 2.19i)T + (-68.5 + 68.5i)T^{2} \)
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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

−9.577731919562279328595716334291, −9.302155891737948508532132297439, −8.469239172368970042790820312992, −7.77265687595167844397556882000, −6.70720720960626561743081979793, −5.44640960594996495815148994044, −5.03869751004363474582193824028, −3.50595961853826023690583165561, −2.43640790108400537271090680921, −0.980923335190066822457301275379, 2.29297696960237962204216173506, 3.39542090303472911026158464850, 3.72019338170232242766038407238, 4.42929881862537723197220192020, 6.77426042036396373416486321105, 7.03174679793050469136370006114, 8.006272209604554870908282416129, 8.854167148672758753078938714901, 9.797020726723181094905589709532, 10.75752196648722042629810030364

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