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Properties

Label 2-3549-3549.2567-c0-0-0
Degree $2$
Conductor $3549$
Sign $-0.963 - 0.266i$
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  + (−0.822 + 0.568i)3-s + (−0.316 − 0.948i)4-s + (−0.0402 − 0.999i)7-s + (0.354 − 0.935i)9-s + (0.799 + 0.600i)12-s + (−0.866 − 0.5i)13-s + (−0.799 + 0.600i)16-s + (−1.14 − 1.14i)19-s + (0.600 + 0.799i)21-s + (−0.960 + 0.278i)25-s + (0.239 + 0.970i)27-s + (−0.935 + 0.354i)28-s + (−0.00243 + 0.120i)31-s + (−0.999 − 0.0402i)36-s + (−0.0296 + 1.46i)37-s + ⋯
L(s)  = 1  + (−0.822 + 0.568i)3-s + (−0.316 − 0.948i)4-s + (−0.0402 − 0.999i)7-s + (0.354 − 0.935i)9-s + (0.799 + 0.600i)12-s + (−0.866 − 0.5i)13-s + (−0.799 + 0.600i)16-s + (−1.14 − 1.14i)19-s + (0.600 + 0.799i)21-s + (−0.960 + 0.278i)25-s + (0.239 + 0.970i)27-s + (−0.935 + 0.354i)28-s + (−0.00243 + 0.120i)31-s + (−0.999 − 0.0402i)36-s + (−0.0296 + 1.46i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $-0.963 - 0.266i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ -0.963 - 0.266i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1967609308\)
\(L(\frac12)\) \(\approx\) \(0.1967609308\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.822 - 0.568i)T \)
7 \( 1 + (0.0402 + 0.999i)T \)
13 \( 1 + (0.866 + 0.5i)T \)
good2 \( 1 + (0.316 + 0.948i)T^{2} \)
5 \( 1 + (0.960 - 0.278i)T^{2} \)
11 \( 1 + (0.663 + 0.748i)T^{2} \)
17 \( 1 + (0.996 + 0.0804i)T^{2} \)
19 \( 1 + (1.14 + 1.14i)T + iT^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.200 - 0.979i)T^{2} \)
31 \( 1 + (0.00243 - 0.120i)T + (-0.999 - 0.0402i)T^{2} \)
37 \( 1 + (0.0296 - 1.46i)T + (-0.999 - 0.0402i)T^{2} \)
41 \( 1 + (-0.160 - 0.987i)T^{2} \)
43 \( 1 + (-1.91 + 0.555i)T + (0.845 - 0.534i)T^{2} \)
47 \( 1 + (-0.391 - 0.919i)T^{2} \)
53 \( 1 + (-0.428 - 0.903i)T^{2} \)
59 \( 1 + (-0.960 + 0.278i)T^{2} \)
61 \( 1 + (0.881 - 1.67i)T + (-0.568 - 0.822i)T^{2} \)
67 \( 1 + (0.103 - 1.70i)T + (-0.992 - 0.120i)T^{2} \)
71 \( 1 + (0.774 - 0.632i)T^{2} \)
73 \( 1 + (-0.0282 - 0.279i)T + (-0.979 + 0.200i)T^{2} \)
79 \( 1 + (0.329 + 1.61i)T + (-0.919 + 0.391i)T^{2} \)
83 \( 1 + (-0.935 + 0.354i)T^{2} \)
89 \( 1 + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (1.91 + 0.271i)T + (0.960 + 0.278i)T^{2} \)
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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

−8.498191022973334296444428138751, −7.35083879079438244597767215772, −6.78468368921784626843707128556, −5.96948046405495118878338054720, −5.30779893455843583954562479470, −4.45681992002294338081762425013, −4.12854198432712548616420180398, −2.72945183626962526252401118502, −1.29027525707338680814299013981, −0.13183505505798889954641291131, 1.94401517376537927148179193876, 2.54388617072536300320993424361, 3.88515985033692632874422015318, 4.57570780194149397590613430267, 5.49576278323008328009567709604, 6.15927831866169871243472559233, 6.90767840084861889045038316712, 7.84613672771288792491811763028, 8.091961551713043744876926353331, 9.163521053144014068995436703613

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