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Properties

Label 2-731-17.16-c1-0-36
Degree $2$
Conductor $731$
Sign $0.117 + 0.993i$
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  − 0.335·2-s − 0.523i·3-s − 1.88·4-s − 1.51i·5-s + 0.175i·6-s + 0.357i·7-s + 1.30·8-s + 2.72·9-s + 0.508i·10-s + 2.34i·11-s + 0.988i·12-s + 2.14·13-s − 0.120i·14-s − 0.792·15-s + 3.33·16-s + (−0.486 − 4.09i)17-s + ⋯
L(s)  = 1  − 0.237·2-s − 0.302i·3-s − 0.943·4-s − 0.676i·5-s + 0.0718i·6-s + 0.135i·7-s + 0.461·8-s + 0.908·9-s + 0.160i·10-s + 0.706i·11-s + 0.285i·12-s + 0.593·13-s − 0.0321i·14-s − 0.204·15-s + 0.833·16-s + (−0.117 − 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.117 + 0.993i$
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: \((2,\ 731,\ (\ :1/2),\ 0.117 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.779515 - 0.692445i\)
\(L(\frac12)\) \(\approx\) \(0.779515 - 0.692445i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (0.486 + 4.09i)T \)
43 \( 1 + T \)
good2 \( 1 + 0.335T + 2T^{2} \)
3 \( 1 + 0.523iT - 3T^{2} \)
5 \( 1 + 1.51iT - 5T^{2} \)
7 \( 1 - 0.357iT - 7T^{2} \)
11 \( 1 - 2.34iT - 11T^{2} \)
13 \( 1 - 2.14T + 13T^{2} \)
19 \( 1 + 4.97T + 19T^{2} \)
23 \( 1 - 0.807iT - 23T^{2} \)
29 \( 1 + 8.18iT - 29T^{2} \)
31 \( 1 + 3.43iT - 31T^{2} \)
37 \( 1 + 10.1iT - 37T^{2} \)
41 \( 1 + 1.18iT - 41T^{2} \)
47 \( 1 + 4.86T + 47T^{2} \)
53 \( 1 + 2.67T + 53T^{2} \)
59 \( 1 - 9.76T + 59T^{2} \)
61 \( 1 + 4.64iT - 61T^{2} \)
67 \( 1 - 2.25T + 67T^{2} \)
71 \( 1 + 2.84iT - 71T^{2} \)
73 \( 1 - 6.73iT - 73T^{2} \)
79 \( 1 - 2.43iT - 79T^{2} \)
83 \( 1 + 0.628T + 83T^{2} \)
89 \( 1 - 0.199T + 89T^{2} \)
97 \( 1 + 4.95iT - 97T^{2} \)
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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

−9.914555445438699332289766981296, −9.357208602679399816987818712064, −8.540961065815854552865227743077, −7.74392075805423513695746584059, −6.81723514541404797206402163127, −5.59426066009105210705986494726, −4.57172277959660947358126807576, −4.00084513739382237719476273556, −2.08963155124395238034416005213, −0.68863664276919989155513020627, 1.37498264958243818143166147641, 3.26741685024306137855387182471, 4.09132959288398748727812387587, 5.02799074479481818492808358589, 6.26974388280750003958656349444, 7.06251932332449839710706489304, 8.353397044566083992679646506801, 8.715473239280287880001153557136, 9.865939616899164732019131758342, 10.56888336779162336109192962424

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