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Properties

Label 2-3675-105.17-c0-0-11
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

\(L(\frac{1}{2})\) \(\approx\) \(0.2729547487\)
\(L(\frac12)\) \(\approx\) \(0.2729547487\)
\(L(1)\) 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

−8.235420932360567160217039367850, −7.28382734715249628142920026629, −6.67360477077056914488199155047, −6.06612478510430540535917873155, −5.32424760639596923963857466637, −4.60434061516885571673223270642, −3.74929538598740793660256393938, −2.65122740493460103353673959177, −1.65425826984806768947031131724, −0.11929427102600731987036377169, 2.17177903320642383545653296540, 3.08866050583678847342942577492, 4.00613964747588150113226680436, 4.44271148828329893864126016998, 5.10379738092124469779527362778, 6.08123673202044193065673290106, 6.47447101277676867969610257528, 7.904545446675540595313049275327, 8.538376628564366041170170093881, 9.091579829332090962583229753570

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