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Properties

Label 2-3675-21.11-c0-0-2
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: $\chi_{3675} (851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ -0.561 - 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8756115635\)
\(L(\frac12)\) \(\approx\) \(0.8756115635\)
\(L(1)\) 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

−8.924414695603105264351206951442, −8.118956776829711920627589841407, −7.81069434476872142991645157290, −7.14634567509054511170173928243, −6.13157864276599480871741306149, −5.44956749822192339412607998883, −4.16342705538322682440497777721, −3.63862956885449887881017342813, −2.41870999014251271829849344511, −1.33551249341891949815630386028, 0.74390691154864697246390050761, 1.81807558903679939290927053020, 2.60062629820537731998628215274, 3.31680615815132261788757386526, 4.43945939322492055934863018037, 5.48779106240820727376157429014, 6.67218608557954941988276258850, 7.18930762590574364945798242277, 8.173179711191154617599816055910, 8.392664380603907987974052453326

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