Properties

Label 2-834-1.1-c1-0-22
Degree $2$
Conductor $834$
Sign $-1$
Analytic cond. $6.65952$
Root an. cond. $2.58060$
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 − 4·5-s + 6-s − 2·7-s + 8-s + 9-s − 4·10-s − 3·11-s + 12-s − 13-s − 2·14-s − 4·15-s + 16-s − 7·17-s + 18-s − 5·19-s − 4·20-s − 2·21-s − 3·22-s + 4·23-s + 24-s + 11·25-s − 26-s + 27-s − 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 1.26·10-s − 0.904·11-s + 0.288·12-s − 0.277·13-s − 0.534·14-s − 1.03·15-s + 1/4·16-s − 1.69·17-s + 0.235·18-s − 1.14·19-s − 0.894·20-s − 0.436·21-s − 0.639·22-s + 0.834·23-s + 0.204·24-s + 11/5·25-s − 0.196·26-s + 0.192·27-s − 0.377·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 834 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 834 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(834\)    =    \(2 \cdot 3 \cdot 139\)
Sign: $-1$
Analytic conductor: \(6.65952\)
Root analytic conductor: \(2.58060\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 834,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 - T \)
139 \( 1 + T \)
good5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−9.864881560234444862713111158936, −8.601309256469704985679713956813, −8.150475819354056643536211398110, −7.06449682939372982450761368402, −6.60305756728717552187460729760, −4.94851772639154862432578069651, −4.24272235212078727358228108207, −3.36373033086053161208519704185, −2.47183382626775911089007214904, 0, 2.47183382626775911089007214904, 3.36373033086053161208519704185, 4.24272235212078727358228108207, 4.94851772639154862432578069651, 6.60305756728717552187460729760, 7.06449682939372982450761368402, 8.150475819354056643536211398110, 8.601309256469704985679713956813, 9.864881560234444862713111158936

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