Properties

Label 2-170-1.1-c1-0-4
Degree $2$
Conductor $170$
Sign $1$
Analytic cond. $1.35745$
Root an. cond. $1.16509$
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 + 3-s + 4-s − 5-s + 6-s + 2·7-s + 8-s − 2·9-s − 10-s + 12-s − 13-s + 2·14-s − 15-s + 16-s − 17-s − 2·18-s − 19-s − 20-s + 2·21-s − 6·23-s + 24-s + 25-s − 26-s − 5·27-s + 2·28-s − 3·29-s − 30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.288·12-s − 0.277·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.471·18-s − 0.229·19-s − 0.223·20-s + 0.436·21-s − 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.962·27-s + 0.377·28-s − 0.557·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $1$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.846901354\)
\(L(\frac12)\) \(\approx\) \(1.846901354\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
17 \( 1 + T \)
good3 \( 1 - T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−12.86758764551411812471231260806, −11.73207427636390486406285519733, −11.16705478363740771468245127663, −9.780585800377850799924183890305, −8.390318499248390643580053193762, −7.75698865717737368803935483355, −6.29665990896415571694545076933, −4.97639963983559004686864229658, −3.77501168661696716981593387272, −2.33045929232568658316872478705, 2.33045929232568658316872478705, 3.77501168661696716981593387272, 4.97639963983559004686864229658, 6.29665990896415571694545076933, 7.75698865717737368803935483355, 8.390318499248390643580053193762, 9.780585800377850799924183890305, 11.16705478363740771468245127663, 11.73207427636390486406285519733, 12.86758764551411812471231260806

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