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Properties

Label 2-3549-3549.2294-c0-0-0
Degree $2$
Conductor $3549$
Sign $0.674 - 0.738i$
Analytic cond. $1.77118$
Root an. cond. $1.33085$
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  + (0.464 + 0.885i)3-s + (0.160 − 0.987i)4-s + (0.692 + 0.721i)7-s + (−0.568 + 0.822i)9-s + (0.948 − 0.316i)12-s + (−0.866 − 0.5i)13-s + (−0.948 − 0.316i)16-s + (0.916 + 0.916i)19-s + (−0.316 + 0.948i)21-s + (0.600 + 0.799i)25-s + (−0.992 − 0.120i)27-s + (0.822 − 0.568i)28-s + (0.761 + 0.306i)31-s + (0.721 + 0.692i)36-s + (1.85 + 0.746i)37-s + ⋯
L(s)  = 1  + (0.464 + 0.885i)3-s + (0.160 − 0.987i)4-s + (0.692 + 0.721i)7-s + (−0.568 + 0.822i)9-s + (0.948 − 0.316i)12-s + (−0.866 − 0.5i)13-s + (−0.948 − 0.316i)16-s + (0.916 + 0.916i)19-s + (−0.316 + 0.948i)21-s + (0.600 + 0.799i)25-s + (−0.992 − 0.120i)27-s + (0.822 − 0.568i)28-s + (0.761 + 0.306i)31-s + (0.721 + 0.692i)36-s + (1.85 + 0.746i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.674 - 0.738i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2294, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.674 - 0.738i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.613255702\)
\(L(\frac12)\) \(\approx\) \(1.613255702\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.464 - 0.885i)T \)
7 \( 1 + (-0.692 - 0.721i)T \)
13 \( 1 + (0.866 + 0.5i)T \)
good2 \( 1 + (-0.160 + 0.987i)T^{2} \)
5 \( 1 + (-0.600 - 0.799i)T^{2} \)
11 \( 1 + (0.935 + 0.354i)T^{2} \)
17 \( 1 + (0.0402 + 0.999i)T^{2} \)
19 \( 1 + (-0.916 - 0.916i)T + iT^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.632 - 0.774i)T^{2} \)
31 \( 1 + (-0.761 - 0.306i)T + (0.721 + 0.692i)T^{2} \)
37 \( 1 + (-1.85 - 0.746i)T + (0.721 + 0.692i)T^{2} \)
41 \( 1 + (-0.0804 + 0.996i)T^{2} \)
43 \( 1 + (-0.866 - 1.15i)T + (-0.278 + 0.960i)T^{2} \)
47 \( 1 + (-0.979 - 0.200i)T^{2} \)
53 \( 1 + (0.845 - 0.534i)T^{2} \)
59 \( 1 + (0.600 + 0.799i)T^{2} \)
61 \( 1 + (0.472 + 1.91i)T + (-0.885 + 0.464i)T^{2} \)
67 \( 1 + (0.646 + 1.43i)T + (-0.663 + 0.748i)T^{2} \)
71 \( 1 + (-0.903 + 0.428i)T^{2} \)
73 \( 1 + (0.600 + 1.68i)T + (-0.774 + 0.632i)T^{2} \)
79 \( 1 + (-0.587 - 0.719i)T + (-0.200 + 0.979i)T^{2} \)
83 \( 1 + (0.822 - 0.568i)T^{2} \)
89 \( 1 + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (0.863 + 1.72i)T + (-0.600 + 0.799i)T^{2} \)
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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

−9.183650561277169076308089418796, −7.987083176266315908804105254455, −7.72708323258029035760149583099, −6.38554127222631961631012582055, −5.65760729148806750177428702343, −4.98400238710327912516324774717, −4.54865788060504408558066432328, −3.18879221336188462075302901706, −2.48517715376733330727875686595, −1.40209848711480406387884617836, 0.982254538114575553949416949738, 2.36201589548378885119570225222, 2.81000434584295294728689694963, 4.03058006220111300504913267586, 4.57356526399606821275849074658, 5.77198851873575314513311730115, 6.87035816326840451823606034482, 7.24395009599816008791674289116, 7.75736576444932256117184724885, 8.497316337800326929615472737634

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