Properties

Label 2-3675-21.2-c0-0-8
Degree $2$
Conductor $3675$
Sign $0.561 - 0.827i$
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  + (1.22 + 0.707i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s − 1.41·6-s + (0.499 − 0.866i)9-s + (−0.866 − 0.5i)12-s + (0.499 − 0.866i)16-s + (1.22 − 0.707i)18-s + (0.707 − 1.22i)19-s + (1.22 + 0.707i)23-s + 0.999i·27-s + (0.707 + 1.22i)31-s + (1.22 − 0.707i)32-s + 0.999·36-s + (1.73 − 0.999i)38-s + ⋯
L(s)  = 1  + (1.22 + 0.707i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s − 1.41·6-s + (0.499 − 0.866i)9-s + (−0.866 − 0.5i)12-s + (0.499 − 0.866i)16-s + (1.22 − 0.707i)18-s + (0.707 − 1.22i)19-s + (1.22 + 0.707i)23-s + 0.999i·27-s + (0.707 + 1.22i)31-s + (1.22 − 0.707i)32-s + 0.999·36-s + (1.73 − 0.999i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.561 - 0.827i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (1451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.561 - 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.950339694\)
\(L(\frac12)\) \(\approx\) \(1.950339694\)
\(L(1)\) 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 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
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

−8.926726042189955947592919412175, −7.69901512224470038991668993902, −6.94492028934407086081256670257, −6.52620535441610807564477718768, −5.65442577967675461031713908475, −4.98360489738100685410801906590, −4.66326067555917149462612004000, −3.59818341518640792518916807816, −2.95358258536124268859835531527, −1.07686141981915975369219622955, 1.18090924044932987349559844839, 2.19911476048663858068875540649, 3.11115047923655715818064812853, 4.08941750954252452125694389242, 4.78610940781188336035139647781, 5.49646999661577311773092131907, 6.07939395807968519555104976083, 6.85140210238901812862380219067, 7.76411305667793230123634270999, 8.406843346066080927369138770288

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