Properties

Label 2-770-1.1-c1-0-9
Degree $2$
Conductor $770$
Sign $1$
Analytic cond. $6.14848$
Root an. cond. $2.47961$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·3-s + 4-s − 5-s − 2·6-s + 7-s − 8-s + 9-s + 10-s − 11-s + 2·12-s + 2·13-s − 14-s − 2·15-s + 16-s + 2·17-s − 18-s + 6·19-s − 20-s + 2·21-s + 22-s + 6·23-s − 2·24-s + 25-s − 2·26-s − 4·27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.577·12-s + 0.554·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.436·21-s + 0.213·22-s + 1.25·23-s − 0.408·24-s + 1/5·25-s − 0.392·26-s − 0.769·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{770} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.630504771\)
\(L(\frac12)\) \(\approx\) \(1.630504771\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−10.10222677050510003671062028867, −9.253271582702521562348222427541, −8.640640021382486068538034692354, −7.80682046403378123605554461083, −7.39856421390674101016288192849, −6.06140846296057676016307473445, −4.82038769623350485466626190745, −3.43318839022200470842040938315, −2.73795458670056035674196653301, −1.22384148583299311271745764464, 1.22384148583299311271745764464, 2.73795458670056035674196653301, 3.43318839022200470842040938315, 4.82038769623350485466626190745, 6.06140846296057676016307473445, 7.39856421390674101016288192849, 7.80682046403378123605554461083, 8.640640021382486068538034692354, 9.253271582702521562348222427541, 10.10222677050510003671062028867

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