Properties

Label 2-735-15.14-c0-0-3
Degree $2$
Conductor $735$
Sign $1$
Analytic cond. $0.366812$
Root an. cond. $0.605650$
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  + 3-s − 4-s + 5-s + 9-s − 12-s + 15-s + 16-s − 2·17-s − 20-s + 25-s + 27-s − 36-s + 45-s − 2·47-s + 48-s − 2·51-s − 60-s − 64-s + 2·68-s + 75-s − 2·79-s + 80-s + 81-s − 2·83-s − 2·85-s − 100-s − 108-s + ⋯
L(s)  = 1  + 3-s − 4-s + 5-s + 9-s − 12-s + 15-s + 16-s − 2·17-s − 20-s + 25-s + 27-s − 36-s + 45-s − 2·47-s + 48-s − 2·51-s − 60-s − 64-s + 2·68-s + 75-s − 2·79-s + 80-s + 81-s − 2·83-s − 2·85-s − 100-s − 108-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.366812\)
Root analytic conductor: \(0.605650\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{735} (344, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.227127003\)
\(L(\frac12)\) \(\approx\) \(1.227127003\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.20278416063923868675093066346, −9.627366540460297318138205177134, −8.842548470518722242354235233287, −8.419387340681146725262179475931, −7.16221285282161101246575092961, −6.21189580242390673333307389569, −4.96090470011521248126898849684, −4.20969157815888785241273240518, −2.94217195339892845878731375317, −1.75562792641656088192317090538, 1.75562792641656088192317090538, 2.94217195339892845878731375317, 4.20969157815888785241273240518, 4.96090470011521248126898849684, 6.21189580242390673333307389569, 7.16221285282161101246575092961, 8.419387340681146725262179475931, 8.842548470518722242354235233287, 9.627366540460297318138205177134, 10.20278416063923868675093066346

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