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Properties

Label 2-735-15.14-c0-0-1
Degree $2$
Conductor $735$
Sign $1$
Analytic cond. $0.366812$
Root an. cond. $0.605650$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s − 3-s + 1.00·4-s + 5-s + 1.41·6-s + 9-s − 1.41·10-s − 1.00·12-s − 15-s − 0.999·16-s − 1.41·18-s − 1.41·19-s + 1.00·20-s + 1.41·23-s + 25-s − 27-s + 1.41·30-s + 1.41·31-s + 1.41·32-s + 1.00·36-s + 2.00·38-s + 45-s − 2.00·46-s + 0.999·48-s − 1.41·50-s + 1.41·53-s + ⋯
L(s)  = 1  − 1.41·2-s − 3-s + 1.00·4-s + 5-s + 1.41·6-s + 9-s − 1.41·10-s − 1.00·12-s − 15-s − 0.999·16-s − 1.41·18-s − 1.41·19-s + 1.00·20-s + 1.41·23-s + 25-s − 27-s + 1.41·30-s + 1.41·31-s + 1.41·32-s + 1.00·36-s + 2.00·38-s + 45-s − 2.00·46-s + 0.999·48-s − 1.41·50-s + 1.41·53-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: no
Arithmetic: yes
Character: $\chi_{735} (344, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4221575161\)
\(L(\frac12)\) \(\approx\) \(0.4221575161\)
\(L(1)\) 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 \( 1 + 1.41T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 1.41T + T^{2} \)
23 \( 1 - 1.41T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.41T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 1.41T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.48767679781548031122155624154, −9.816805548434013102121918236944, −9.032920328246445280370002573946, −8.220174536731727986824345885605, −6.97633822972332313412119909893, −6.50765901467596082400325519957, −5.39579545452888574929467137370, −4.41019616365952847647548395155, −2.36507033203502314396116263133, −1.11613813786054256982614528162, 1.11613813786054256982614528162, 2.36507033203502314396116263133, 4.41019616365952847647548395155, 5.39579545452888574929467137370, 6.50765901467596082400325519957, 6.97633822972332313412119909893, 8.220174536731727986824345885605, 9.032920328246445280370002573946, 9.816805548434013102121918236944, 10.48767679781548031122155624154

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