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Properties

Label 2-675-3.2-c0-0-1
Degree $2$
Conductor $675$
Sign $1$
Analytic cond. $0.336868$
Root an. cond. $0.580404$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 7-s + 2·13-s + 16-s − 19-s − 28-s − 31-s − 37-s − 43-s + 2·52-s − 61-s + 64-s + 2·67-s − 73-s − 76-s − 79-s − 2·91-s − 97-s − 103-s − 109-s − 112-s + ⋯
L(s)  = 1  + 4-s − 7-s + 2·13-s + 16-s − 19-s − 28-s − 31-s − 37-s − 43-s + 2·52-s − 61-s + 64-s + 2·67-s − 73-s − 76-s − 79-s − 2·91-s − 97-s − 103-s − 109-s − 112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.336868\)
Root analytic conductor: \(0.580404\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{675} (26, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.102988900\)
\(L(\frac12)\) \(\approx\) \(1.102988900\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T + T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T + T^{2} \)
67 \( ( 1 - T )^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( 1 + T + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.80486013552528133144479930558, −10.02139774674197987110527100148, −8.913454287815251210423209095739, −8.156647047220134200149927456496, −6.92443973789230618453641744316, −6.37809467847737027682363330410, −5.60504957090055148819803596068, −3.89219645812292862097592955568, −3.11757965444193515574870143045, −1.69075851728213846931389449038, 1.69075851728213846931389449038, 3.11757965444193515574870143045, 3.89219645812292862097592955568, 5.60504957090055148819803596068, 6.37809467847737027682363330410, 6.92443973789230618453641744316, 8.156647047220134200149927456496, 8.913454287815251210423209095739, 10.02139774674197987110527100148, 10.80486013552528133144479930558

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