Properties

Label 2-7800-1.1-c1-0-111
Degree $2$
Conductor $7800$
Sign $-1$
Analytic cond. $62.2833$
Root an. cond. $7.89197$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3.35·7-s + 9-s − 3.81·11-s − 13-s − 17-s + 4.24·19-s + 3.35·21-s − 6.24·23-s + 27-s − 6.78·29-s − 5.97·31-s − 3.81·33-s − 4.45·37-s − 39-s − 0.621·41-s + 1.54·43-s − 9.14·47-s + 4.24·49-s − 51-s − 7.08·53-s + 4.24·57-s + 0.272·59-s + 4.78·61-s + 3.35·63-s + 1.81·67-s − 6.24·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.26·7-s + 0.333·9-s − 1.14·11-s − 0.277·13-s − 0.242·17-s + 0.974·19-s + 0.731·21-s − 1.30·23-s + 0.192·27-s − 1.26·29-s − 1.07·31-s − 0.663·33-s − 0.733·37-s − 0.160·39-s − 0.0971·41-s + 0.234·43-s − 1.33·47-s + 0.606·49-s − 0.140·51-s − 0.973·53-s + 0.562·57-s + 0.0354·59-s + 0.613·61-s + 0.422·63-s + 0.221·67-s − 0.752·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7800\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(62.2833\)
Root analytic conductor: \(7.89197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
13 \( 1 + T \)
good7 \( 1 - 3.35T + 7T^{2} \)
11 \( 1 + 3.81T + 11T^{2} \)
17 \( 1 + T + 17T^{2} \)
19 \( 1 - 4.24T + 19T^{2} \)
23 \( 1 + 6.24T + 23T^{2} \)
29 \( 1 + 6.78T + 29T^{2} \)
31 \( 1 + 5.97T + 31T^{2} \)
37 \( 1 + 4.45T + 37T^{2} \)
41 \( 1 + 0.621T + 41T^{2} \)
43 \( 1 - 1.54T + 43T^{2} \)
47 \( 1 + 9.14T + 47T^{2} \)
53 \( 1 + 7.08T + 53T^{2} \)
59 \( 1 - 0.272T + 59T^{2} \)
61 \( 1 - 4.78T + 61T^{2} \)
67 \( 1 - 1.81T + 67T^{2} \)
71 \( 1 + 9.16T + 71T^{2} \)
73 \( 1 + 3.32T + 73T^{2} \)
79 \( 1 + 5.54T + 79T^{2} \)
83 \( 1 + 6.27T + 83T^{2} \)
89 \( 1 - 3.78T + 89T^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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

−7.59348669089643492903855164420, −7.17673928439489633514636083625, −5.97566201735451679822998892963, −5.28768935166878612199687951321, −4.77176406187070853417758546222, −3.88802356237877723893602003907, −3.08393306460702167811867346805, −2.11311745575661065775572778395, −1.58157505950349790360055074143, 0, 1.58157505950349790360055074143, 2.11311745575661065775572778395, 3.08393306460702167811867346805, 3.88802356237877723893602003907, 4.77176406187070853417758546222, 5.28768935166878612199687951321, 5.97566201735451679822998892963, 7.17673928439489633514636083625, 7.59348669089643492903855164420

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