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Properties

Label 1-3549-3549.2687-r0-0-0
Degree $1$
Conductor $3549$
Sign $-0.389 - 0.921i$
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  + (0.919 − 0.391i)2-s + (0.692 − 0.721i)4-s + (0.885 − 0.464i)5-s + (0.354 − 0.935i)8-s + (0.632 − 0.774i)10-s + (−0.799 + 0.600i)11-s + (−0.0402 − 0.999i)16-s + (−0.632 − 0.774i)17-s + (0.5 − 0.866i)19-s + (0.278 − 0.960i)20-s + (−0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + (0.568 − 0.822i)25-s + (0.919 − 0.391i)29-s + (−0.568 − 0.822i)31-s + (−0.428 − 0.903i)32-s + ⋯
L(s)  = 1  + (0.919 − 0.391i)2-s + (0.692 − 0.721i)4-s + (0.885 − 0.464i)5-s + (0.354 − 0.935i)8-s + (0.632 − 0.774i)10-s + (−0.799 + 0.600i)11-s + (−0.0402 − 0.999i)16-s + (−0.632 − 0.774i)17-s + (0.5 − 0.866i)19-s + (0.278 − 0.960i)20-s + (−0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + (0.568 − 0.822i)25-s + (0.919 − 0.391i)29-s + (−0.568 − 0.822i)31-s + (−0.428 − 0.903i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.389 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.389 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $-0.389 - 0.921i$
Analytic conductor: \(16.4814\)
Root analytic conductor: \(16.4814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2687, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (0:\ ),\ -0.389 - 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.963145778 - 2.960340263i\)
\(L(\frac12)\) \(\approx\) \(1.963145778 - 2.960340263i\)
\(L(1)\) \(\approx\) \(1.804819982 - 0.9652372249i\)
\(L(1)\) \(\approx\) \(1.804819982 - 0.9652372249i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.919 - 0.391i)T \)
5 \( 1 + (0.885 - 0.464i)T \)
11 \( 1 + (-0.799 + 0.600i)T \)
17 \( 1 + (-0.632 - 0.774i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.919 - 0.391i)T \)
31 \( 1 + (-0.568 - 0.822i)T \)
37 \( 1 + (0.428 - 0.903i)T \)
41 \( 1 + (0.948 + 0.316i)T \)
43 \( 1 + (0.428 + 0.903i)T \)
47 \( 1 + (-0.970 + 0.239i)T \)
53 \( 1 + (0.354 - 0.935i)T \)
59 \( 1 + (-0.0402 + 0.999i)T \)
61 \( 1 + (-0.987 - 0.160i)T \)
67 \( 1 + (0.692 + 0.721i)T \)
71 \( 1 + (0.200 + 0.979i)T \)
73 \( 1 + (-0.120 - 0.992i)T \)
79 \( 1 + (-0.970 + 0.239i)T \)
83 \( 1 + (-0.748 - 0.663i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.845 - 0.534i)T \)
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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

−18.76374094959997845136141859706, −18.20785630125953643383254733723, −17.431976810500857185791205782301, −16.78073954400850495115036786130, −16.11342310983146772785053617185, −15.41218401292933174011994917548, −14.66933244775237810491533971894, −14.06680843879355438218468512792, −13.57017899902707967107818578676, −12.77069292381541805356028380595, −12.34107098219462927919025193979, −11.1727533159577043997102721661, −10.71048351674260831527115909896, −10.07594169944503076358381005877, −8.90665468188813623119568490776, −8.27427050807680302547997988213, −7.42809324596131436489167867931, −6.5955904595366759050014893551, −6.09245250976088790465150980477, −5.3663978608706706227924637117, −4.71818074411037171670599110318, −3.66445846835878252666452580065, −2.94482208409532626366490753761, −2.29786411517199362917676747178, −1.34736594410304038255513159196, 0.684132864417183916582015865221, 1.66440473928562891688263693811, 2.516201229820193833488032136731, 2.94534496370504134847586719104, 4.28632584738512444442607600753, 4.788259025791858567211186061650, 5.47280816627416312197935901664, 6.11176225493682142673909322241, 7.043170933608489659445576118, 7.627904846139424160681907388896, 8.89078244309440359404409175612, 9.62261730898815686372713219447, 10.032611152806763186498265133947, 11.08715327850364703572891893607, 11.47822771960001429081318805401, 12.51741701360545605749127044342, 13.09719382315682335488272419576, 13.445870895571549075617987555786, 14.23116021154088167464242197853, 14.92639790627114763478836754166, 15.821038787736181055901497366, 16.10244528945467571364330005685, 17.18850885071836975302729235880, 17.93085985045377246109048901925, 18.361331331579966764571346914163

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