Properties

Label 2-188760-1.1-c1-0-26
Degree $2$
Conductor $188760$
Sign $1$
Analytic cond. $1507.25$
Root an. cond. $38.8233$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 4·7-s + 9-s + 13-s + 15-s + 6·17-s + 4·19-s − 4·21-s + 8·23-s + 25-s + 27-s − 4·35-s + 12·37-s + 39-s − 10·41-s − 4·43-s + 45-s + 6·47-s + 9·49-s + 6·51-s + 6·53-s + 4·57-s − 4·59-s − 6·61-s − 4·63-s + 65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s + 1.45·17-s + 0.917·19-s − 0.872·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.676·35-s + 1.97·37-s + 0.160·39-s − 1.56·41-s − 0.609·43-s + 0.149·45-s + 0.875·47-s + 9/7·49-s + 0.840·51-s + 0.824·53-s + 0.529·57-s − 0.520·59-s − 0.768·61-s − 0.503·63-s + 0.124·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(188760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(1507.25\)
Root analytic conductor: \(38.8233\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{188760} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 188760,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.100824922\)
\(L(\frac12)\) \(\approx\) \(4.100824922\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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

−13.15121986791660, −12.70701970483265, −12.41826052107886, −11.65640815137387, −11.37804526645908, −10.53451641335073, −10.04942681431086, −9.915082485098090, −9.340243942506406, −8.909400663253500, −8.544757297333528, −7.683015583791633, −7.374882283522126, −6.941852847582767, −6.179617006236826, −6.020663776081855, −5.280623137377194, −4.840744066195990, −4.027873506410966, −3.409154886662468, −3.042742834145199, −2.782689747571903, −1.846595949141486, −1.095886593539659, −0.6292043843658601, 0.6292043843658601, 1.095886593539659, 1.846595949141486, 2.782689747571903, 3.042742834145199, 3.409154886662468, 4.027873506410966, 4.840744066195990, 5.280623137377194, 6.020663776081855, 6.179617006236826, 6.941852847582767, 7.374882283522126, 7.683015583791633, 8.544757297333528, 8.909400663253500, 9.340243942506406, 9.915082485098090, 10.04942681431086, 10.53451641335073, 11.37804526645908, 11.65640815137387, 12.41826052107886, 12.70701970483265, 13.15121986791660

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