Properties

Label 2-722-1.1-c3-0-65
Degree $2$
Conductor $722$
Sign $-1$
Analytic cond. $42.5993$
Root an. cond. $6.52681$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 5.78·3-s + 4·4-s + 6.19·5-s − 11.5·6-s + 11.2·7-s + 8·8-s + 6.45·9-s + 12.3·10-s − 45.1·11-s − 23.1·12-s + 22.6·13-s + 22.4·14-s − 35.8·15-s + 16·16-s − 31.0·17-s + 12.9·18-s + 24.7·20-s − 64.9·21-s − 90.2·22-s − 144.·23-s − 46.2·24-s − 86.5·25-s + 45.3·26-s + 118.·27-s + 44.8·28-s + 24.2·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.11·3-s + 0.5·4-s + 0.554·5-s − 0.787·6-s + 0.605·7-s + 0.353·8-s + 0.238·9-s + 0.392·10-s − 1.23·11-s − 0.556·12-s + 0.483·13-s + 0.428·14-s − 0.617·15-s + 0.250·16-s − 0.442·17-s + 0.168·18-s + 0.277·20-s − 0.674·21-s − 0.874·22-s − 1.30·23-s − 0.393·24-s − 0.692·25-s + 0.341·26-s + 0.847·27-s + 0.302·28-s + 0.155·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(42.5993\)
Root analytic conductor: \(6.52681\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 722,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
19 \( 1 \)
good3 \( 1 + 5.78T + 27T^{2} \)
5 \( 1 - 6.19T + 125T^{2} \)
7 \( 1 - 11.2T + 343T^{2} \)
11 \( 1 + 45.1T + 1.33e3T^{2} \)
13 \( 1 - 22.6T + 2.19e3T^{2} \)
17 \( 1 + 31.0T + 4.91e3T^{2} \)
23 \( 1 + 144.T + 1.21e4T^{2} \)
29 \( 1 - 24.2T + 2.43e4T^{2} \)
31 \( 1 - 303.T + 2.97e4T^{2} \)
37 \( 1 + 121.T + 5.06e4T^{2} \)
41 \( 1 + 103.T + 6.89e4T^{2} \)
43 \( 1 + 444.T + 7.95e4T^{2} \)
47 \( 1 - 481.T + 1.03e5T^{2} \)
53 \( 1 + 441.T + 1.48e5T^{2} \)
59 \( 1 - 242.T + 2.05e5T^{2} \)
61 \( 1 + 531.T + 2.26e5T^{2} \)
67 \( 1 + 727.T + 3.00e5T^{2} \)
71 \( 1 + 225.T + 3.57e5T^{2} \)
73 \( 1 + 351.T + 3.89e5T^{2} \)
79 \( 1 + 602.T + 4.93e5T^{2} \)
83 \( 1 - 248.T + 5.71e5T^{2} \)
89 \( 1 + 1.66e3T + 7.04e5T^{2} \)
97 \( 1 + 846.T + 9.12e5T^{2} \)
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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

−10.09133564842582289687928095550, −8.547604128990597816323783369619, −7.74717930590740554736459708580, −6.49584707758645639499914935171, −5.88389821390692879701814521091, −5.14600474574963263525811320098, −4.36705200307461310774753661785, −2.83621882268981337765840736320, −1.63835454561042631940280827935, 0, 1.63835454561042631940280827935, 2.83621882268981337765840736320, 4.36705200307461310774753661785, 5.14600474574963263525811320098, 5.88389821390692879701814521091, 6.49584707758645639499914935171, 7.74717930590740554736459708580, 8.547604128990597816323783369619, 10.09133564842582289687928095550

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