Properties

Label 2-735-1.1-c1-0-5
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 4-s − 5-s − 6-s − 3·8-s + 9-s − 10-s + 12-s + 6·13-s + 15-s − 16-s − 2·17-s + 18-s + 8·19-s + 20-s + 8·23-s + 3·24-s + 25-s + 6·26-s − 27-s − 2·29-s + 30-s − 4·31-s + 5·32-s − 2·34-s − 36-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 1.66·13-s + 0.258·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 1.83·19-s + 0.223·20-s + 1.66·23-s + 0.612·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s − 0.371·29-s + 0.182·30-s − 0.718·31-s + 0.883·32-s − 0.342·34-s − 1/6·36-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: $\chi_{735} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.437375972\)
\(L(\frac12)\) \(\approx\) \(1.437375972\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−10.66117211895074937453677481475, −9.357583073556529155557794013065, −8.843078054411908501657519136744, −7.69164818258356184893378968692, −6.67009631001529545506185276841, −5.69208011630684437322158261218, −5.01564976002989168468776414543, −3.94068846921577588199342611811, −3.18167275817234212778601509206, −0.982469269817440515290174005672, 0.982469269817440515290174005672, 3.18167275817234212778601509206, 3.94068846921577588199342611811, 5.01564976002989168468776414543, 5.69208011630684437322158261218, 6.67009631001529545506185276841, 7.69164818258356184893378968692, 8.843078054411908501657519136744, 9.357583073556529155557794013065, 10.66117211895074937453677481475

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