Properties

Label 2-845-1.1-c1-0-12
Degree $2$
Conductor $845$
Sign $1$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s − 4-s + 5-s − 2·6-s + 4·7-s − 3·8-s + 9-s + 10-s − 2·11-s + 2·12-s + 4·14-s − 2·15-s − 16-s + 2·17-s + 18-s + 6·19-s − 20-s − 8·21-s − 2·22-s − 6·23-s + 6·24-s + 25-s + 4·27-s − 4·28-s + 2·29-s − 2·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.447·5-s − 0.816·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.577·12-s + 1.06·14-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 − 1.74·21-s − 0.426·22-s − 1.25·23-s + 1.22·24-s + 1/5·25-s + 0.769·27-s − 0.755·28-s + 0.371·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.492934938\)
\(L(\frac12)\) \(\approx\) \(1.492934938\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
13 \( 1 \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.34630410671677379936192033935, −9.532805961208409547451766916679, −8.353551523811345329615757792465, −7.69003698049592494171610769456, −6.24360515405325801186702466344, −5.56389466213996946942866194317, −5.01117336241618809550332242437, −4.27130437726285993901225335150, −2.71745145781960607840515064653, −0.996154674950302563103745756297, 0.996154674950302563103745756297, 2.71745145781960607840515064653, 4.27130437726285993901225335150, 5.01117336241618809550332242437, 5.56389466213996946942866194317, 6.24360515405325801186702466344, 7.69003698049592494171610769456, 8.353551523811345329615757792465, 9.532805961208409547451766916679, 10.34630410671677379936192033935

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