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Properties

Label 2-3675-105.68-c0-0-3
Degree $2$
Conductor $3675$
Sign $-0.193 - 0.981i$
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  + (0.965 + 0.258i)3-s + (−0.866 + 0.5i)4-s + (0.866 + 0.499i)9-s + (−0.965 + 0.258i)12-s + (−0.707 + 0.707i)13-s + (0.499 − 0.866i)16-s + (−0.866 + 1.5i)19-s + (0.707 + 0.707i)27-s − 36-s + (0.448 + 1.67i)37-s + (−0.866 + 0.500i)39-s + (0.707 − 0.707i)48-s + (0.258 − 0.965i)52-s + (−1.22 + 1.22i)57-s + (1.5 + 0.866i)61-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)3-s + (−0.866 + 0.5i)4-s + (0.866 + 0.499i)9-s + (−0.965 + 0.258i)12-s + (−0.707 + 0.707i)13-s + (0.499 − 0.866i)16-s + (−0.866 + 1.5i)19-s + (0.707 + 0.707i)27-s − 36-s + (0.448 + 1.67i)37-s + (−0.866 + 0.500i)39-s + (0.707 − 0.707i)48-s + (0.258 − 0.965i)52-s + (−1.22 + 1.22i)57-s + (1.5 + 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-0.193 - 0.981i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ -0.193 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.263706957\)
\(L(\frac12)\) \(\approx\) \(1.263706957\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
17 \( 1 + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
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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

−8.712884685191409536985375974854, −8.371048602186860739496296743718, −7.65356145733996450840416983453, −6.94697749189251308125167624211, −5.88189506198397345035536874259, −4.78478164368481981431789141137, −4.28991555720789405356216645152, −3.54808640542475688959872520020, −2.68100766912975153095555963227, −1.61503739749990397508887998196, 0.66422596783113207668556116391, 2.04147857753441681615122307736, 2.87647462385648734453166396351, 3.91804962965775805254257061152, 4.60217595397053385247377368807, 5.38067579058960162890240395404, 6.33631070581481351965226516245, 7.18031035216958064346832685309, 7.85799027563540742231004120300, 8.656765066294601026614341664694

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