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Properties

Label 2-731-17.16-c1-0-20
Degree $2$
Conductor $731$
Sign $-0.665 - 0.746i$
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 + 2.30i·3-s + 0.995·4-s + 0.771i·5-s + 3.99i·6-s + 3.40i·7-s − 1.73·8-s − 2.31·9-s + 1.33i·10-s − 1.19i·11-s + 2.29i·12-s + 3.11·13-s + 5.89i·14-s − 1.77·15-s − 4.99·16-s + (−2.74 − 3.07i)17-s + ⋯
L(s)  = 1  + 1.22·2-s + 1.33i·3-s + 0.497·4-s + 0.345i·5-s + 1.62i·6-s + 1.28i·7-s − 0.614·8-s − 0.772·9-s + 0.422i·10-s − 0.360i·11-s + 0.662i·12-s + 0.864·13-s + 1.57i·14-s − 0.459·15-s − 1.24·16-s + (−0.665 − 0.746i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.665 - 0.746i$
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: \((2,\ 731,\ (\ :1/2),\ -0.665 - 0.746i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.993066 + 2.21571i\)
\(L(\frac12)\) \(\approx\) \(0.993066 + 2.21571i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (2.74 + 3.07i)T \)
43 \( 1 - T \)
good2 \( 1 - 1.73T + 2T^{2} \)
3 \( 1 - 2.30iT - 3T^{2} \)
5 \( 1 - 0.771iT - 5T^{2} \)
7 \( 1 - 3.40iT - 7T^{2} \)
11 \( 1 + 1.19iT - 11T^{2} \)
13 \( 1 - 3.11T + 13T^{2} \)
19 \( 1 + 1.18T + 19T^{2} \)
23 \( 1 - 1.80iT - 23T^{2} \)
29 \( 1 - 3.79iT - 29T^{2} \)
31 \( 1 + 4.18iT - 31T^{2} \)
37 \( 1 - 0.0262iT - 37T^{2} \)
41 \( 1 - 10.8iT - 41T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 + 4.81T + 53T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 + 8.94iT - 61T^{2} \)
67 \( 1 - 9.61T + 67T^{2} \)
71 \( 1 - 14.8iT - 71T^{2} \)
73 \( 1 + 5.92iT - 73T^{2} \)
79 \( 1 + 7.15iT - 79T^{2} \)
83 \( 1 - 4.12T + 83T^{2} \)
89 \( 1 + 2.65T + 89T^{2} \)
97 \( 1 + 8.58iT - 97T^{2} \)
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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

−11.08544261516863883015363393408, −9.785868917845304487812583351278, −9.068238339788503542505200409279, −8.485821039349786134765564582365, −6.76441318434110575315325360486, −5.84821961351856645725367959870, −5.18041505873629969261929753488, −4.34747611586793535418119156811, −3.40468325757137022010489367805, −2.60206587114847194751206417903, 0.874693325670024834039791263698, 2.26210221233507519993415868174, 3.76997645628419188679248081804, 4.41726160025347725931830170842, 5.64201088457488693642631594576, 6.61420979787382135336522079644, 7.05475089593270626480144946833, 8.186603934311482181169748046362, 8.966847266444552157251615508863, 10.39529024714366773866282765071

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