Properties

Label 2-735-105.74-c0-0-3
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: $\chi_{735} (704, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :0),\ 0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.267208684\)
\(L(\frac12)\) \(\approx\) \(1.267208684\)
\(L(1)\) 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

−10.52637397281877366073093152534, −9.689503881927085556120129940704, −8.742794392041753860315530280391, −7.75555311219985220427541705270, −6.64341832279841336983118199946, −6.43537031924491084636755520879, −5.40760113346433075488120997423, −3.71211874767177296529468498933, −2.43476684438956917686842830607, −1.68298445597528661425097146136, 2.17800736411101080810521990340, 3.13171993246734502106979498647, 4.37078128411502001449074164289, 5.02964532871386960084274842880, 6.30718773024176410848288525851, 7.45699921777825010921990131947, 8.282295216607599097573046521155, 9.108103775034608040782763722498, 9.571897958238636919640746391690, 10.73943095168790138141127992420

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