Properties

Label 2-731-17.16-c1-0-2
Degree $2$
Conductor $731$
Sign $-0.320 + 0.947i$
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  + 3.20i·3-s − 2·4-s + 0.837i·5-s + 2.87i·7-s − 7.25·9-s − 6.13i·11-s − 6.40i·12-s − 4.57·13-s − 2.68·15-s + 4·16-s + (−1.31 + 3.90i)17-s + 2·19-s − 1.67i·20-s − 9.21·21-s + 5.48i·23-s + ⋯
L(s)  = 1  + 1.84i·3-s − 4-s + 0.374i·5-s + 1.08i·7-s − 2.41·9-s − 1.85i·11-s − 1.84i·12-s − 1.26·13-s − 0.692·15-s + 16-s + (−0.320 + 0.947i)17-s + 0.458·19-s − 0.374i·20-s − 2.00·21-s + 1.14i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.320 + 0.947i$
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: \((2,\ 731,\ (\ :1/2),\ -0.320 + 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.236888 - 0.330082i\)
\(L(\frac12)\) \(\approx\) \(0.236888 - 0.330082i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (1.31 - 3.90i)T \)
43 \( 1 - T \)
good2 \( 1 + 2T^{2} \)
3 \( 1 - 3.20iT - 3T^{2} \)
5 \( 1 - 0.837iT - 5T^{2} \)
7 \( 1 - 2.87iT - 7T^{2} \)
11 \( 1 + 6.13iT - 11T^{2} \)
13 \( 1 + 4.57T + 13T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 5.48iT - 23T^{2} \)
29 \( 1 + 1.67iT - 29T^{2} \)
31 \( 1 + 1.40iT - 31T^{2} \)
37 \( 1 + 3.85iT - 37T^{2} \)
41 \( 1 - 5.11iT - 41T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + 2.82T + 53T^{2} \)
59 \( 1 + 7.29T + 59T^{2} \)
61 \( 1 + 9.90iT - 61T^{2} \)
67 \( 1 + 8.73T + 67T^{2} \)
71 \( 1 + 8.76iT - 71T^{2} \)
73 \( 1 - 15.6iT - 73T^{2} \)
79 \( 1 - 8.56iT - 79T^{2} \)
83 \( 1 + 1.52T + 83T^{2} \)
89 \( 1 - 3.05T + 89T^{2} \)
97 \( 1 + 7.16iT - 97T^{2} \)
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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

−10.90572667224160288491426696172, −9.898728039645114916239344838174, −9.389710641114898435228870660769, −8.691430590675312311093511296991, −8.052625779825755521412080100449, −6.04664006006771936282346847293, −5.44523419300502771090311120182, −4.69513562021482626557725425641, −3.56328137257221388213857688324, −2.92865930797399720515730928875, 0.22131400347888335048552973460, 1.45371145472481927430521357821, 2.77110844290525030950734551725, 4.54627747059582986621185878634, 5.07070403292974928962161616436, 6.64634383287841302336075498837, 7.28103814390627964068188713534, 7.74542922762109005204407670309, 8.827021303044983350686348556274, 9.664499992912570769560882735074

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