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Properties

Label 2-3549-3549.3086-c0-0-0
Degree $2$
Conductor $3549$
Sign $0.999 - 0.0278i$
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  + (−0.992 + 0.120i)3-s + (−0.464 − 0.885i)4-s + (−0.748 − 0.663i)7-s + (0.970 − 0.239i)9-s + (0.568 + 0.822i)12-s + i·13-s + (−0.568 + 0.822i)16-s + (1.21 + 1.21i)19-s + (0.822 + 0.568i)21-s + (−0.935 − 0.354i)25-s + (−0.935 + 0.354i)27-s + (−0.239 + 0.970i)28-s + (−0.783 + 1.74i)31-s + (−0.663 − 0.748i)36-s + (0.819 − 1.82i)37-s + ⋯
L(s)  = 1  + (−0.992 + 0.120i)3-s + (−0.464 − 0.885i)4-s + (−0.748 − 0.663i)7-s + (0.970 − 0.239i)9-s + (0.568 + 0.822i)12-s + i·13-s + (−0.568 + 0.822i)16-s + (1.21 + 1.21i)19-s + (0.822 + 0.568i)21-s + (−0.935 − 0.354i)25-s + (−0.935 + 0.354i)27-s + (−0.239 + 0.970i)28-s + (−0.783 + 1.74i)31-s + (−0.663 − 0.748i)36-s + (0.819 − 1.82i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.999 - 0.0278i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (3086, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.999 - 0.0278i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6540907012\)
\(L(\frac12)\) \(\approx\) \(0.6540907012\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.992 - 0.120i)T \)
7 \( 1 + (0.748 + 0.663i)T \)
13 \( 1 - iT \)
good2 \( 1 + (0.464 + 0.885i)T^{2} \)
5 \( 1 + (0.935 + 0.354i)T^{2} \)
11 \( 1 + (0.464 - 0.885i)T^{2} \)
17 \( 1 + (-0.120 + 0.992i)T^{2} \)
19 \( 1 + (-1.21 - 1.21i)T + iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.885 + 0.464i)T^{2} \)
31 \( 1 + (0.783 - 1.74i)T + (-0.663 - 0.748i)T^{2} \)
37 \( 1 + (-0.819 + 1.82i)T + (-0.663 - 0.748i)T^{2} \)
41 \( 1 + (0.239 + 0.970i)T^{2} \)
43 \( 1 + (-1.24 - 0.470i)T + (0.748 + 0.663i)T^{2} \)
47 \( 1 + (-0.822 + 0.568i)T^{2} \)
53 \( 1 + (-0.120 + 0.992i)T^{2} \)
59 \( 1 + (-0.935 - 0.354i)T^{2} \)
61 \( 1 + (-1.17 + 1.32i)T + (-0.120 - 0.992i)T^{2} \)
67 \( 1 + (0.783 - 0.244i)T + (0.822 - 0.568i)T^{2} \)
71 \( 1 + (-0.239 - 0.970i)T^{2} \)
73 \( 1 + (-0.308 + 0.186i)T + (0.464 - 0.885i)T^{2} \)
79 \( 1 + (-1.75 - 0.922i)T + (0.568 + 0.822i)T^{2} \)
83 \( 1 + (-0.239 + 0.970i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (-1.39 + 0.254i)T + (0.935 - 0.354i)T^{2} \)
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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

−9.194397614997553423728371481133, −7.82065442553830269333357469341, −7.08377701235675520226549838881, −6.36500001986672187531429145721, −5.77038110351472429516214509088, −5.08980329419513477849833538327, −4.14840635942018495927369423277, −3.64207929267415976687383425763, −1.89005789977290521152542000989, −0.876957218499768739678577921333, 0.62665274310650836540833534781, 2.41186969178390479910648430697, 3.25982330230133860300127043933, 4.14661596120691278108370211758, 5.09167734952586092363480018521, 5.67967598600944851984978542227, 6.43442598976005904974856774243, 7.42728385062901491792113640001, 7.73924139398036802608930198171, 8.810098219537307711706329350723

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