Properties

Label 2-735-1.1-c1-0-23
Degree $2$
Conductor $735$
Sign $-1$
Analytic cond. $5.86900$
Root an. cond. $2.42260$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s + 9-s − 2·10-s − 6·11-s + 2·12-s − 3·13-s + 15-s − 4·16-s − 4·17-s − 2·18-s + 19-s + 2·20-s + 12·22-s − 4·23-s + 25-s + 6·26-s + 27-s − 8·29-s − 2·30-s + 31-s + 8·32-s − 6·33-s + 8·34-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 1/3·9-s − 0.632·10-s − 1.80·11-s + 0.577·12-s − 0.832·13-s + 0.258·15-s − 16-s − 0.970·17-s − 0.471·18-s + 0.229·19-s + 0.447·20-s + 2.55·22-s − 0.834·23-s + 1/5·25-s + 1.17·26-s + 0.192·27-s − 1.48·29-s − 0.365·30-s + 0.179·31-s + 1.41·32-s − 1.04·33-s + 1.37·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−9.748395957074705728171348053572, −9.245180260331893745789230775379, −8.190905413721799078782891747604, −7.73500063744226361021680192017, −6.88801796507275225595285396641, −5.54104550542340361135724955036, −4.45568069611345949977480819854, −2.71878989525937820035970356719, −1.94009500889479360979174607930, 0, 1.94009500889479360979174607930, 2.71878989525937820035970356719, 4.45568069611345949977480819854, 5.54104550542340361135724955036, 6.88801796507275225595285396641, 7.73500063744226361021680192017, 8.190905413721799078782891747604, 9.245180260331893745789230775379, 9.748395957074705728171348053572

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