Properties

Label 2-731-17.16-c1-0-31
Degree $2$
Conductor $731$
Sign $0.969 + 0.244i$
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  − 1.52·2-s + 1.41i·3-s + 0.337·4-s − 1.28i·5-s − 2.15i·6-s + 2.51i·7-s + 2.54·8-s + 1.00·9-s + 1.96i·10-s − 5.00i·11-s + 0.476i·12-s − 0.374·13-s − 3.85i·14-s + 1.81·15-s − 4.56·16-s + (−3.99 − 1.00i)17-s + ⋯
L(s)  = 1  − 1.08·2-s + 0.814i·3-s + 0.168·4-s − 0.574i·5-s − 0.880i·6-s + 0.952i·7-s + 0.898·8-s + 0.335·9-s + 0.621i·10-s − 1.50i·11-s + 0.137i·12-s − 0.104·13-s − 1.02i·14-s + 0.468·15-s − 1.14·16-s + (−0.969 − 0.244i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.969 + 0.244i$
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: \((2,\ 731,\ (\ :1/2),\ 0.969 + 0.244i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.757964 - 0.0939260i\)
\(L(\frac12)\) \(\approx\) \(0.757964 - 0.0939260i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (3.99 + 1.00i)T \)
43 \( 1 + T \)
good2 \( 1 + 1.52T + 2T^{2} \)
3 \( 1 - 1.41iT - 3T^{2} \)
5 \( 1 + 1.28iT - 5T^{2} \)
7 \( 1 - 2.51iT - 7T^{2} \)
11 \( 1 + 5.00iT - 11T^{2} \)
13 \( 1 + 0.374T + 13T^{2} \)
19 \( 1 - 1.22T + 19T^{2} \)
23 \( 1 + 5.61iT - 23T^{2} \)
29 \( 1 + 0.871iT - 29T^{2} \)
31 \( 1 + 7.21iT - 31T^{2} \)
37 \( 1 - 2.29iT - 37T^{2} \)
41 \( 1 + 4.11iT - 41T^{2} \)
47 \( 1 - 10.6T + 47T^{2} \)
53 \( 1 - 7.42T + 53T^{2} \)
59 \( 1 - 3.05T + 59T^{2} \)
61 \( 1 + 1.29iT - 61T^{2} \)
67 \( 1 - 7.23T + 67T^{2} \)
71 \( 1 - 0.777iT - 71T^{2} \)
73 \( 1 - 11.2iT - 73T^{2} \)
79 \( 1 + 6.45iT - 79T^{2} \)
83 \( 1 + 5.26T + 83T^{2} \)
89 \( 1 - 4.39T + 89T^{2} \)
97 \( 1 + 8.20iT - 97T^{2} \)
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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

−10.21001761347829236916221064866, −9.259108993553481499966043383357, −8.816632336396842988350434639034, −8.305145007691224815977398838308, −7.06233964302501230524696718905, −5.81563088845801477920298056480, −4.87263282208402967994282312104, −3.98707653060489058092296016150, −2.43839250857528871918173776359, −0.70540207157620533975104702944, 1.14618926872901180718443614114, 2.17173150687958875188481154443, 3.95929152946399716909989259838, 4.88789635052140549696773718630, 6.64696247434579175806259535426, 7.26880354060379659208992199945, 7.51625496410029698059428874867, 8.717574604655742155188138385490, 9.647416009613293118307236375245, 10.31469555990879536273765893384

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