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Properties

Label 2-731-17.15-c1-0-61
Degree $2$
Conductor $731$
Sign $0.739 - 0.673i$
Analytic cond. $5.83706$
Root an. cond. $2.41600$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.292 − 0.292i)2-s + (−1 − 2.41i)3-s − 1.82i·4-s + (−3.41 + 1.41i)5-s + (−0.414 + i)6-s + (−2.41 − i)7-s + (−1.12 + 1.12i)8-s + (−2.70 + 2.70i)9-s + (1.41 + 0.585i)10-s + (0.292 − 0.707i)11-s + (−4.41 + 1.82i)12-s − 1.17i·13-s + (0.414 + i)14-s + (6.82 + 6.82i)15-s − 3·16-s + (4.12 − 0.121i)17-s + ⋯
L(s)  = 1  + (−0.207 − 0.207i)2-s + (−0.577 − 1.39i)3-s − 0.914i·4-s + (−1.52 + 0.632i)5-s + (−0.169 + 0.408i)6-s + (−0.912 − 0.377i)7-s + (−0.396 + 0.396i)8-s + (−0.902 + 0.902i)9-s + (0.447 + 0.185i)10-s + (0.0883 − 0.213i)11-s + (−1.27 + 0.527i)12-s − 0.324i·13-s + (0.110 + 0.267i)14-s + (1.76 + 1.76i)15-s − 0.750·16-s + (0.999 − 0.0294i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.739 - 0.673i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (474, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.739 - 0.673i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-4.12 + 0.121i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (0.292 + 0.292i)T + 2iT^{2} \)
3 \( 1 + (1 + 2.41i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (3.41 - 1.41i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (2.41 + i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.292 + 0.707i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 1.17iT - 13T^{2} \)
19 \( 1 + (2 + 2i)T + 19iT^{2} \)
23 \( 1 + (-0.121 + 0.292i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (0.878 + 2.12i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (-2.24 - 5.41i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (6.12 + 2.53i)T + (28.9 + 28.9i)T^{2} \)
47 \( 1 + 0.242iT - 47T^{2} \)
53 \( 1 + (-5.82 - 5.82i)T + 53iT^{2} \)
59 \( 1 + (-9.48 + 9.48i)T - 59iT^{2} \)
61 \( 1 + (-3.41 - 1.41i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 7.07T + 67T^{2} \)
71 \( 1 + (-6.07 - 14.6i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (12.2 - 5.07i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-2.46 + 5.94i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (1.17 + 1.17i)T + 83iT^{2} \)
89 \( 1 - 9.17iT - 89T^{2} \)
97 \( 1 + (11.7 - 4.87i)T + (68.5 - 68.5i)T^{2} \)
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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

−9.906354205584478769871232498296, −8.561323579277210338341437183644, −7.67992152744237982596454702904, −6.90857399730255261500977813133, −6.39893166291528719343550040615, −5.35709781759557708417602410168, −3.85241020044506007706365457691, −2.67906135022428548548399029787, −0.982909405266385198144487805545, 0, 3.21804298140366788022005569640, 3.86713323988229477778237157841, 4.54002746304117085643394825441, 5.66288876354892142787340630204, 6.89911745963119817772951027748, 7.84754813051447201204018049580, 8.673618699971264163317054918110, 9.351269187242196173510677182913, 10.18895330364864617106688779959

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