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Properties

Label 2-704-44.43-c1-0-21
Degree $2$
Conductor $704$
Sign $-i$
Analytic cond. $5.62146$
Root an. cond. $2.37096$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.31i·3-s − 3·5-s − 8·9-s + 3.31i·11-s + 9.94i·15-s + 3.31i·23-s + 4·25-s + 16.5i·27-s − 9.94i·31-s + 11·33-s − 7·37-s + 24·45-s + 6.63i·47-s − 7·49-s − 6·53-s + ⋯
L(s)  = 1  − 1.91i·3-s − 1.34·5-s − 2.66·9-s + 1.00i·11-s + 2.56i·15-s + 0.691i·23-s + 0.800·25-s + 3.19i·27-s − 1.78i·31-s + 1.91·33-s − 1.15·37-s + 3.57·45-s + 0.967i·47-s − 49-s − 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $-i$
Analytic conductor: \(5.62146\)
Root analytic conductor: \(2.37096\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{704} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 704,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 - 3.31iT \)
good3 \( 1 + 3.31iT - 3T^{2} \)
5 \( 1 + 3T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 3.31iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 9.94iT - 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 6.63iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 3.31iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 9.94iT - 67T^{2} \)
71 \( 1 - 16.5iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + 17T + 97T^{2} \)
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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

−9.666790338973587515622523736203, −8.493572644889958533464086128990, −7.81490559442887454751120629388, −7.31493305562632928867511194204, −6.58221105065033544939748795416, −5.45762465322230015236764744954, −4.07743882859935420998233066208, −2.78435280141425654658403708045, −1.54249487062163749925713422612, 0, 3.12698146552960151806648300996, 3.64305100300231984201598075692, 4.60097662653138778358261583646, 5.33641654692689145409582078387, 6.57670977616043792829215727312, 8.035784411417860809709606041602, 8.594361729608586267554812323925, 9.326312862081773032875803090117, 10.49894472254340161155107142913

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