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Properties

Label 2-735-105.44-c0-0-1
Degree $2$
Conductor $735$
Sign $0.605 - 0.795i$
Analytic cond. $0.366812$
Root an. cond. $0.605650$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.499 + 0.866i)9-s + (0.499 − 0.866i)12-s + 0.999·15-s + (−0.499 + 0.866i)16-s + (1 + 1.73i)17-s − 0.999·20-s + (−0.499 − 0.866i)25-s + 0.999·27-s − 0.999·36-s + (−0.499 − 0.866i)45-s + (1 − 1.73i)47-s + 0.999·48-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.499 + 0.866i)9-s + (0.499 − 0.866i)12-s + 0.999·15-s + (−0.499 + 0.866i)16-s + (1 + 1.73i)17-s − 0.999·20-s + (−0.499 − 0.866i)25-s + 0.999·27-s − 0.999·36-s + (−0.499 − 0.866i)45-s + (1 − 1.73i)47-s + 0.999·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(0.366812\)
Root analytic conductor: \(0.605650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :0),\ 0.605 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8023067512\)
\(L(\frac12)\) \(\approx\) \(0.8023067512\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + 2T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−10.83204794524187949345945510961, −10.26492124758546436534391457664, −8.572525401937391288519209964904, −7.934911789847468180717148483804, −7.25587218916819884663728333365, −6.50394563203440409360598921695, −5.69355142421212428121645499423, −4.05548203707898481199449903525, −3.10105258152074799171612639044, −1.91756844726631483147636055121, 0.966931755699272803534671108655, 2.91723167025477944060945751034, 4.26803873455902628686651272357, 5.15218043123087279590044835958, 5.69177334164751718093501738474, 6.87321358368359352618730766245, 7.86644752114061662478791488124, 9.160262943298433047611678621537, 9.569763323085022777656256631319, 10.45874606832548006004769144146

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