↑

Properties

Label 2-7350-1.1-c1-0-98
Degree $2$
Conductor $7350$
Sign $-1$
Analytic cond. $58.6900$
Root an. cond. $7.66094$
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 − 8-s + 9-s + 12-s + 5·13-s + 16-s − 6·17-s − 18-s − 7·19-s − 6·23-s − 24-s − 5·26-s + 27-s + 8·31-s − 32-s + 6·34-s + 36-s − 37-s + 7·38-s + 5·39-s + 8·43-s + 6·46-s + 6·47-s + 48-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 1.38·13-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 1.60·19-s − 1.25·23-s − 0.204·24-s − 0.980·26-s + 0.192·27-s + 1.43·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.164·37-s + 1.13·38-s + 0.800·39-s + 1.21·43-s + 0.884·46-s + 0.875·47-s + 0.144·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(58.6900\)
Root analytic conductor: \(7.66094\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7350,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 + 18 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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

−7.86564772399101540812472942018, −6.79621731348569378955799005828, −6.39693522399184540369684821868, −5.73529041650560217373285066358, −4.28890470329402967238204990918, −4.12750714798830684104245206691, −2.89270249179781840968460027158, −2.19149611071272922865251218271, −1.35021743478435357980701340718, 0, 1.35021743478435357980701340718, 2.19149611071272922865251218271, 2.89270249179781840968460027158, 4.12750714798830684104245206691, 4.28890470329402967238204990918, 5.73529041650560217373285066358, 6.39693522399184540369684821868, 6.79621731348569378955799005828, 7.86564772399101540812472942018

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