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Properties

Label 2-731-17.16-c1-0-48
Degree $2$
Conductor $731$
Sign $-0.922 + 0.387i$
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  − 1.27·2-s − 0.638i·3-s − 0.363·4-s − 1.64i·5-s + 0.816i·6-s − 4.36i·7-s + 3.02·8-s + 2.59·9-s + 2.10i·10-s − 4.08i·11-s + 0.231i·12-s − 7.00·13-s + 5.58i·14-s − 1.04·15-s − 3.14·16-s + (3.80 − 1.59i)17-s + ⋯
L(s)  = 1  − 0.904·2-s − 0.368i·3-s − 0.181·4-s − 0.735i·5-s + 0.333i·6-s − 1.64i·7-s + 1.06·8-s + 0.864·9-s + 0.665i·10-s − 1.23i·11-s + 0.0669i·12-s − 1.94·13-s + 1.49i·14-s − 0.270·15-s − 0.785·16-s + (0.922 − 0.387i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.922 + 0.387i$
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: \((2,\ 731,\ (\ :1/2),\ -0.922 + 0.387i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.144476 - 0.717530i\)
\(L(\frac12)\) \(\approx\) \(0.144476 - 0.717530i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-3.80 + 1.59i)T \)
43 \( 1 + T \)
good2 \( 1 + 1.27T + 2T^{2} \)
3 \( 1 + 0.638iT - 3T^{2} \)
5 \( 1 + 1.64iT - 5T^{2} \)
7 \( 1 + 4.36iT - 7T^{2} \)
11 \( 1 + 4.08iT - 11T^{2} \)
13 \( 1 + 7.00T + 13T^{2} \)
19 \( 1 - 5.02T + 19T^{2} \)
23 \( 1 - 5.58iT - 23T^{2} \)
29 \( 1 - 3.44iT - 29T^{2} \)
31 \( 1 + 1.85iT - 31T^{2} \)
37 \( 1 + 2.73iT - 37T^{2} \)
41 \( 1 + 6.36iT - 41T^{2} \)
47 \( 1 + 8.92T + 47T^{2} \)
53 \( 1 + 6.95T + 53T^{2} \)
59 \( 1 - 3.93T + 59T^{2} \)
61 \( 1 - 11.5iT - 61T^{2} \)
67 \( 1 - 1.27T + 67T^{2} \)
71 \( 1 + 10.6iT - 71T^{2} \)
73 \( 1 - 14.0iT - 73T^{2} \)
79 \( 1 + 9.62iT - 79T^{2} \)
83 \( 1 + 4.62T + 83T^{2} \)
89 \( 1 + 5.75T + 89T^{2} \)
97 \( 1 - 7.14iT - 97T^{2} \)
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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

−9.844110414382721916573142820927, −9.381376406865539547093999500031, −8.171307328916435613110627213483, −7.35086029022669099338842476976, −7.21778931715422991091167550434, −5.32946993421362001161397317189, −4.58550522690296751617182774634, −3.44962571518505930690961407322, −1.36190923667307148444799186087, −0.58381345270227968933196802667, 1.87305807047440692008029228779, 2.94696913563924314548256950212, 4.65415902396677839306213939368, 5.13104398960788668751854187956, 6.66751347553645180699416006819, 7.46821643063529977762368401166, 8.219176238621503510323732085867, 9.429519332278487034605650368319, 9.801339195314677666098385227509, 10.21553082417106536752636209318

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