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Properties

Label 1-3549-3549.101-r0-0-0
Degree $1$
Conductor $3549$
Sign $-0.926 - 0.377i$
Analytic cond. $16.4814$
Root an. cond. $16.4814$
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  + (−0.200 + 0.979i)2-s + (−0.919 − 0.391i)4-s + (−0.692 + 0.721i)5-s + (0.568 − 0.822i)8-s + (−0.568 − 0.822i)10-s + (−0.748 + 0.663i)11-s + (0.692 + 0.721i)16-s + (0.428 + 0.903i)17-s + 19-s + (0.919 − 0.391i)20-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.0402 − 0.999i)25-s + (−0.948 + 0.316i)29-s + (−0.845 − 0.534i)31-s + (−0.845 + 0.534i)32-s + ⋯
L(s)  = 1  + (−0.200 + 0.979i)2-s + (−0.919 − 0.391i)4-s + (−0.692 + 0.721i)5-s + (0.568 − 0.822i)8-s + (−0.568 − 0.822i)10-s + (−0.748 + 0.663i)11-s + (0.692 + 0.721i)16-s + (0.428 + 0.903i)17-s + 19-s + (0.919 − 0.391i)20-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.0402 − 0.999i)25-s + (−0.948 + 0.316i)29-s + (−0.845 − 0.534i)31-s + (−0.845 + 0.534i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $-0.926 - 0.377i$
Analytic conductor: \(16.4814\)
Root analytic conductor: \(16.4814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (0:\ ),\ -0.926 - 0.377i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1598936944 + 0.8159835509i\)
\(L(\frac12)\) \(\approx\) \(-0.1598936944 + 0.8159835509i\)
\(L(1)\) \(\approx\) \(0.5435649884 + 0.5055987422i\)
\(L(1)\) \(\approx\) \(0.5435649884 + 0.5055987422i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.200 + 0.979i)T \)
5 \( 1 + (-0.692 + 0.721i)T \)
11 \( 1 + (-0.748 + 0.663i)T \)
17 \( 1 + (0.428 + 0.903i)T \)
19 \( 1 + T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.948 + 0.316i)T \)
31 \( 1 + (-0.845 - 0.534i)T \)
37 \( 1 + (0.845 + 0.534i)T \)
41 \( 1 + (0.632 + 0.774i)T \)
43 \( 1 + (-0.0402 - 0.999i)T \)
47 \( 1 + (-0.799 - 0.600i)T \)
53 \( 1 + (0.996 + 0.0804i)T \)
59 \( 1 + (-0.692 + 0.721i)T \)
61 \( 1 + (-0.568 - 0.822i)T \)
67 \( 1 + (-0.120 + 0.992i)T \)
71 \( 1 + (0.987 - 0.160i)T \)
73 \( 1 + (0.948 + 0.316i)T \)
79 \( 1 + (0.799 + 0.600i)T \)
83 \( 1 + (0.354 + 0.935i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.692 + 0.721i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.40247443410043389682656767946, −17.99661794455609579947445904203, −16.83637328164495224672848588384, −16.422626364119064621735302727979, −15.81131122260758302279702863640, −14.76208591709696153302881100081, −14.00724559152661504618801082277, −13.25845982077700430486734346579, −12.68771820014283637024237494709, −12.0766743248287347634479171376, −11.2388211622817482257297914930, −10.95021363975611400778286151347, −9.88137282554928101452789900997, −9.210113783781094483714846840521, −8.67042514723446987110857580748, −7.678783617542742789887974893499, −7.500382644915133725178332454548, −5.91949254712949843076624529274, −5.097032683374581072639625726159, −4.63635121438046648501645877294, −3.56357194704446863271751792861, −3.09754924930762774111207927447, −2.12200491067561743140398560851, −1.00848373452997775154860613063, −0.359163377751118110628435109270, 1.0359832428436748785657194061, 2.20184517771876507121553143179, 3.39823087657266147548895984973, 3.90090011889995228151293291576, 4.947293009495529830614768301621, 5.554782682651183810459359823019, 6.40390835559473213763491142621, 7.27344193002325647304698361754, 7.63510310359499791036175932554, 8.23166856189938959620487617377, 9.29243301963610609371946618185, 9.872681366252413380766827501443, 10.64595265770190024268080572299, 11.32643644388992606735413171245, 12.294050784053499136056203950841, 13.03979921455162482442179583950, 13.69699644002673246602309341399, 14.6156708330289635410051469062, 15.10306902072680279952446095327, 15.485685805026188237839425932889, 16.373758848150072613646019363989, 16.90314884112904190825554199511, 17.835486274503352583328316179359, 18.36921643935852691020350247379, 18.79433634294744246107623570608

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