↑

Properties

Label 2-127050-1.1-c1-0-164
Degree $2$
Conductor $127050$
Sign $-1$
Analytic cond. $1014.49$
Root an. cond. $31.8512$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 12-s + 3·13-s + 14-s + 16-s + 6·17-s − 18-s − 3·19-s + 21-s + 5·23-s + 24-s − 3·26-s − 27-s − 28-s + 7·29-s − 3·31-s − 32-s − 6·34-s + 36-s + 4·37-s + 3·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.832·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.688·19-s + 0.218·21-s + 1.04·23-s + 0.204·24-s − 0.588·26-s − 0.192·27-s − 0.188·28-s + 1.29·29-s − 0.538·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.657·37-s + 0.486·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(127050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(1014.49\)
Root analytic conductor: \(31.8512\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 127050,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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

−13.80284120574594, −13.01277004601094, −12.67593656785927, −12.41345577301964, −11.60782768720079, −11.30399244824081, −10.86335327380958, −10.32167265834460, −9.869671409553850, −9.520816682196967, −8.816921114313993, −8.416660856875744, −7.864322731373745, −7.395824955939986, −6.722584483990560, −6.300661276781700, −5.997120665625018, −5.146037967411668, −4.875659740254610, −3.931091832185640, −3.436962600319567, −2.881680606645166, −2.123004792652735, −1.238490107256300, −0.9300675549455507, 0, 0.9300675549455507, 1.238490107256300, 2.123004792652735, 2.881680606645166, 3.436962600319567, 3.931091832185640, 4.875659740254610, 5.146037967411668, 5.997120665625018, 6.300661276781700, 6.722584483990560, 7.395824955939986, 7.864322731373745, 8.416660856875744, 8.816921114313993, 9.520816682196967, 9.869671409553850, 10.32167265834460, 10.86335327380958, 11.30399244824081, 11.60782768720079, 12.41345577301964, 12.67593656785927, 13.01277004601094, 13.80284120574594

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