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Properties

Label 2-3675-105.44-c0-0-2
Degree $2$
Conductor $3675$
Sign $-0.441 - 0.897i$
Analytic cond. $1.83406$
Root an. cond. $1.35427$
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  + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (0.499 − 0.866i)9-s + (−0.866 − 0.499i)12-s + i·13-s + (−0.499 + 0.866i)16-s + (0.5 − 0.866i)19-s + 0.999i·27-s + (1 + 1.73i)31-s + 0.999·36-s + (−0.866 − 0.5i)37-s + (−0.5 − 0.866i)39-s + 2i·43-s − 0.999i·48-s + (−0.866 + 0.5i)52-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (0.499 − 0.866i)9-s + (−0.866 − 0.499i)12-s + i·13-s + (−0.499 + 0.866i)16-s + (0.5 − 0.866i)19-s + 0.999i·27-s + (1 + 1.73i)31-s + 0.999·36-s + (−0.866 − 0.5i)37-s + (−0.5 − 0.866i)39-s + 2i·43-s − 0.999i·48-s + (−0.866 + 0.5i)52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-0.441 - 0.897i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (2774, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ -0.441 - 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9859213073\)
\(L(\frac12)\) \(\approx\) \(0.9859213073\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - iT - T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 2iT - T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + iT - T^{2} \)
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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

−8.984767087519394013858337831238, −8.246527043961474102848346115722, −7.21253464960143543298550965015, −6.77880033856685841147471674128, −6.12057600771092268005644362242, −5.02764445749980491801074010514, −4.44999246186176198179348033402, −3.56070869084224192559535995863, −2.73657932803332693568753137616, −1.42156983889669302031478567055, 0.66246478570059125016810864412, 1.71567785294318915552831204670, 2.64389787921711101402887745290, 3.91130683483380075218868320430, 5.07587083301006102929593675467, 5.52187759384646165642743226718, 6.19194121835197976168501001974, 6.84418602362574487617387465763, 7.64698468749278219105202131006, 8.211653793453758075988065845458

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