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Properties

Label 2-3675-105.17-c0-0-6
Degree $2$
Conductor $3675$
Sign $0.886 + 0.463i$
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  + (−0.739 − 0.198i)2-s + (0.258 + 0.965i)3-s + (−0.358 − 0.207i)4-s − 0.765i·6-s + (0.765 + 0.765i)8-s + (−0.866 + 0.499i)9-s + (0.107 − 0.400i)12-s + (−0.207 − 0.358i)16-s + (1.36 − 0.366i)17-s + (0.739 − 0.198i)18-s + (−0.923 − 1.60i)19-s + (0.478 − 1.78i)23-s + (−0.541 + 0.937i)24-s + (−0.707 − 0.707i)27-s + (−0.662 − 0.382i)31-s + (−0.198 − 0.739i)32-s + ⋯
L(s)  = 1  + (−0.739 − 0.198i)2-s + (0.258 + 0.965i)3-s + (−0.358 − 0.207i)4-s − 0.765i·6-s + (0.765 + 0.765i)8-s + (−0.866 + 0.499i)9-s + (0.107 − 0.400i)12-s + (−0.207 − 0.358i)16-s + (1.36 − 0.366i)17-s + (0.739 − 0.198i)18-s + (−0.923 − 1.60i)19-s + (0.478 − 1.78i)23-s + (−0.541 + 0.937i)24-s + (−0.707 − 0.707i)27-s + (−0.662 − 0.382i)31-s + (−0.198 − 0.739i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.886 + 0.463i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (2432, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.886 + 0.463i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7313192749\)
\(L(\frac12)\) \(\approx\) \(0.7313192749\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.739 + 0.198i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.478 + 1.78i)T + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.662 + 0.382i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (-1.78 + 0.478i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1 + i)T - iT^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - iT^{2} \)
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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

−8.964917094516853034246466304913, −8.190033041380439701842022884233, −7.48763883120659480663623092496, −6.41464145570439617463419172524, −5.42950412504619624368789864581, −4.77976478799509204272761213738, −4.18901292235419663759434413442, −3.02748773988327209221175664932, −2.21097190314850953137301541814, −0.63118368622050488775657415732, 1.12923670489695540489361418254, 1.92891121449169736804913691623, 3.42437596566061519245804007640, 3.80023118830152581902868290678, 5.25604326314324099475816481738, 5.88746542007205790965858341930, 6.85052291059466717619543431474, 7.53832551230328957032486250381, 7.991730902019293123117882119018, 8.593880458413078770037051921137

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