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Properties

Label 2-731-17.16-c1-0-23
Degree $2$
Conductor $731$
Sign $0.293 - 0.955i$
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  − 1.16·2-s + 0.590i·3-s − 0.636·4-s + 3.35i·5-s − 0.689i·6-s − 0.173i·7-s + 3.07·8-s + 2.65·9-s − 3.91i·10-s − 2.50i·11-s − 0.376i·12-s + 3.55·13-s + 0.202i·14-s − 1.97·15-s − 2.32·16-s + (1.21 − 3.94i)17-s + ⋯
L(s)  = 1  − 0.825·2-s + 0.341i·3-s − 0.318·4-s + 1.49i·5-s − 0.281i·6-s − 0.0656i·7-s + 1.08·8-s + 0.883·9-s − 1.23i·10-s − 0.755i·11-s − 0.108i·12-s + 0.986·13-s + 0.0542i·14-s − 0.511·15-s − 0.580·16-s + (0.293 − 0.955i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.293 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.293 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.293 - 0.955i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (560, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.293 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.808310 + 0.597055i\)
\(L(\frac12)\) \(\approx\) \(0.808310 + 0.597055i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-1.21 + 3.94i)T \)
43 \( 1 - T \)
good2 \( 1 + 1.16T + 2T^{2} \)
3 \( 1 - 0.590iT - 3T^{2} \)
5 \( 1 - 3.35iT - 5T^{2} \)
7 \( 1 + 0.173iT - 7T^{2} \)
11 \( 1 + 2.50iT - 11T^{2} \)
13 \( 1 - 3.55T + 13T^{2} \)
19 \( 1 - 5.87T + 19T^{2} \)
23 \( 1 - 5.88iT - 23T^{2} \)
29 \( 1 - 0.661iT - 29T^{2} \)
31 \( 1 + 3.16iT - 31T^{2} \)
37 \( 1 + 0.381iT - 37T^{2} \)
41 \( 1 - 10.7iT - 41T^{2} \)
47 \( 1 + 1.79T + 47T^{2} \)
53 \( 1 - 3.08T + 53T^{2} \)
59 \( 1 + 9.68T + 59T^{2} \)
61 \( 1 + 2.07iT - 61T^{2} \)
67 \( 1 + 7.02T + 67T^{2} \)
71 \( 1 + 8.13iT - 71T^{2} \)
73 \( 1 - 0.992iT - 73T^{2} \)
79 \( 1 - 3.44iT - 79T^{2} \)
83 \( 1 - 2.85T + 83T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 + 5.87iT - 97T^{2} \)
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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

−10.45063805540951027230719460630, −9.678160947483490479810518536089, −9.153058479461902495274510102995, −7.80198320499530815198651728338, −7.40118880523526189411034090277, −6.34811399761717646024623988286, −5.19062843749814152616398060981, −3.86580332683298120528903834920, −3.07692597383564176988810838581, −1.24454402884996920958134574810, 0.920935015862791925328383415831, 1.69646059309603370226056883985, 3.96144647747040549955223014129, 4.67775603025165292308127181782, 5.66394023574224460903331079421, 7.02829042180444261817699137402, 7.88387040704186709263566213203, 8.606355999498561759122678102427, 9.198702188111456153348649294047, 10.03043068033242559679089437469

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