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Properties

Label 2-3549-3549.2582-c0-0-1
Degree $2$
Conductor $3549$
Sign $0.613 - 0.790i$
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  + (−0.239 + 0.970i)3-s + (−0.822 + 0.568i)4-s + (0.120 − 0.992i)7-s + (−0.885 − 0.464i)9-s + (−0.354 − 0.935i)12-s − i·13-s + (0.354 − 0.935i)16-s + (−1.35 + 1.35i)19-s + (0.935 + 0.354i)21-s + (0.663 + 0.748i)25-s + (0.663 − 0.748i)27-s + (0.464 + 0.885i)28-s + (1.96 − 0.118i)31-s + (0.992 − 0.120i)36-s + (1.57 − 0.0950i)37-s + ⋯
L(s)  = 1  + (−0.239 + 0.970i)3-s + (−0.822 + 0.568i)4-s + (0.120 − 0.992i)7-s + (−0.885 − 0.464i)9-s + (−0.354 − 0.935i)12-s − i·13-s + (0.354 − 0.935i)16-s + (−1.35 + 1.35i)19-s + (0.935 + 0.354i)21-s + (0.663 + 0.748i)25-s + (0.663 − 0.748i)27-s + (0.464 + 0.885i)28-s + (1.96 − 0.118i)31-s + (0.992 − 0.120i)36-s + (1.57 − 0.0950i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.613 - 0.790i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2582, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.613 - 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8778186065\)
\(L(\frac12)\) \(\approx\) \(0.8778186065\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.239 - 0.970i)T \)
7 \( 1 + (-0.120 + 0.992i)T \)
13 \( 1 + iT \)
good2 \( 1 + (0.822 - 0.568i)T^{2} \)
5 \( 1 + (-0.663 - 0.748i)T^{2} \)
11 \( 1 + (0.822 + 0.568i)T^{2} \)
17 \( 1 + (0.970 - 0.239i)T^{2} \)
19 \( 1 + (1.35 - 1.35i)T - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.568 - 0.822i)T^{2} \)
31 \( 1 + (-1.96 + 0.118i)T + (0.992 - 0.120i)T^{2} \)
37 \( 1 + (-1.57 + 0.0950i)T + (0.992 - 0.120i)T^{2} \)
41 \( 1 + (-0.464 + 0.885i)T^{2} \)
43 \( 1 + (-1.31 - 1.48i)T + (-0.120 + 0.992i)T^{2} \)
47 \( 1 + (-0.935 + 0.354i)T^{2} \)
53 \( 1 + (0.970 - 0.239i)T^{2} \)
59 \( 1 + (0.663 + 0.748i)T^{2} \)
61 \( 1 + (1.12 + 0.136i)T + (0.970 + 0.239i)T^{2} \)
67 \( 1 + (-1.96 + 0.359i)T + (0.935 - 0.354i)T^{2} \)
71 \( 1 + (0.464 - 0.885i)T^{2} \)
73 \( 1 + (1.74 + 0.542i)T + (0.822 + 0.568i)T^{2} \)
79 \( 1 + (-0.271 + 0.393i)T + (-0.354 - 0.935i)T^{2} \)
83 \( 1 + (0.464 + 0.885i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (-0.580 - 1.28i)T + (-0.663 + 0.748i)T^{2} \)
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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

−8.872792165373739519343515320174, −8.011075245434573790100022282582, −7.80214210123137545556691739193, −6.45498694738094339425171867774, −5.77948767159412000876260847248, −4.70429197821812624689443166728, −4.36038846850897131293465301974, −3.56191786457498262137951015716, −2.80504652016588137418556178477, −0.862059983734764813922707177666, 0.803195381995767767048060133083, 2.08819860682818991335222190297, 2.72077059788733947596231848899, 4.36624069637822193304486217250, 4.78732630887810007283785693968, 5.82050508473388692059863530529, 6.32765018866214221430564240040, 6.96386426945945406179489336433, 8.099377864565459651974672710883, 8.721769921791365764697583803593

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