Properties

Label 2-722-1.1-c1-0-26
Degree $2$
Conductor $722$
Sign $-1$
Analytic cond. $5.76519$
Root an. cond. $2.40108$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s − 2·9-s − 6·11-s − 12-s − 5·13-s − 14-s + 16-s + 3·17-s − 2·18-s + 21-s − 6·22-s + 3·23-s − 24-s − 5·25-s − 5·26-s + 5·27-s − 28-s − 9·29-s + 4·31-s + 32-s + 6·33-s + 3·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s − 1.80·11-s − 0.288·12-s − 1.38·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.471·18-s + 0.218·21-s − 1.27·22-s + 0.625·23-s − 0.204·24-s − 25-s − 0.980·26-s + 0.962·27-s − 0.188·28-s − 1.67·29-s + 0.718·31-s + 0.176·32-s + 1.04·33-s + 0.514·34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
19 \( 1 \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−10.22778802623068126404197182867, −9.303239213765892386789415426985, −7.87022552956968366194501622566, −7.40476303111443747598071967849, −6.09727019472392450239856120896, −5.41480429146700181457700017195, −4.76456060038860011441331940251, −3.24158791522542483628555695808, −2.38005475536401603079615670633, 0, 2.38005475536401603079615670633, 3.24158791522542483628555695808, 4.76456060038860011441331940251, 5.41480429146700181457700017195, 6.09727019472392450239856120896, 7.40476303111443747598071967849, 7.87022552956968366194501622566, 9.303239213765892386789415426985, 10.22778802623068126404197182867

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