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Properties

Label 2-3549-3549.2552-c0-0-0
Degree $2$
Conductor $3549$
Sign $0.140 + 0.990i$
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  + (0.632 − 0.774i)3-s + (−0.120 + 0.992i)4-s + (−0.692 − 0.721i)7-s + (−0.200 − 0.979i)9-s + (0.692 + 0.721i)12-s + (0.5 − 0.866i)13-s + (−0.970 − 0.239i)16-s + (0.548 + 0.316i)19-s + (−0.996 + 0.0804i)21-s + (0.0402 − 0.999i)25-s + (−0.885 − 0.464i)27-s + (0.799 − 0.600i)28-s + (−1.06 − 1.67i)31-s + (0.996 − 0.0804i)36-s + (0.0747 − 0.142i)37-s + ⋯
L(s)  = 1  + (0.632 − 0.774i)3-s + (−0.120 + 0.992i)4-s + (−0.692 − 0.721i)7-s + (−0.200 − 0.979i)9-s + (0.692 + 0.721i)12-s + (0.5 − 0.866i)13-s + (−0.970 − 0.239i)16-s + (0.548 + 0.316i)19-s + (−0.996 + 0.0804i)21-s + (0.0402 − 0.999i)25-s + (−0.885 − 0.464i)27-s + (0.799 − 0.600i)28-s + (−1.06 − 1.67i)31-s + (0.996 − 0.0804i)36-s + (0.0747 − 0.142i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.140 + 0.990i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2552, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.140 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.281494211\)
\(L(\frac12)\) \(\approx\) \(1.281494211\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.632 + 0.774i)T \)
7 \( 1 + (0.692 + 0.721i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.120 - 0.992i)T^{2} \)
5 \( 1 + (-0.0402 + 0.999i)T^{2} \)
11 \( 1 + (-0.919 - 0.391i)T^{2} \)
17 \( 1 + (0.354 - 0.935i)T^{2} \)
19 \( 1 + (-0.548 - 0.316i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.919 - 0.391i)T^{2} \)
31 \( 1 + (1.06 + 1.67i)T + (-0.428 + 0.903i)T^{2} \)
37 \( 1 + (-0.0747 + 0.142i)T + (-0.568 - 0.822i)T^{2} \)
41 \( 1 + (0.948 + 0.316i)T^{2} \)
43 \( 1 + (-1.68 - 1.06i)T + (0.428 + 0.903i)T^{2} \)
47 \( 1 + (0.278 - 0.960i)T^{2} \)
53 \( 1 + (-0.987 + 0.160i)T^{2} \)
59 \( 1 + (0.885 - 0.464i)T^{2} \)
61 \( 1 + (0.685 + 1.44i)T + (-0.632 + 0.774i)T^{2} \)
67 \( 1 + (-0.558 - 0.743i)T + (-0.278 + 0.960i)T^{2} \)
71 \( 1 + (-0.200 - 0.979i)T^{2} \)
73 \( 1 + (0.608 + 1.82i)T + (-0.799 + 0.600i)T^{2} \)
79 \( 1 + (-0.566 + 0.426i)T + (0.278 - 0.960i)T^{2} \)
83 \( 1 + (-0.748 - 0.663i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-1.66 + 0.481i)T + (0.845 - 0.534i)T^{2} \)
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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

−8.392497823733688098768595230043, −7.59640824950220913551309598629, −7.53937696150208837375634543931, −6.43106704485149038752779559201, −5.88816186959468245696879417533, −4.44858758736610808679321742808, −3.65899196174080742924005322401, −3.12056516404612877702609107688, −2.20780000510249164310897452377, −0.69907683671738862888816654867, 1.53876190640361520842187296351, 2.53135263049672521673897805625, 3.46216072357941818241003696039, 4.28424651502850178247594258277, 5.27890710212487949910392887838, 5.64918241127267011192758303379, 6.67243401977620885875295438820, 7.34990569833494412045226486840, 8.596464101225107127000551245987, 9.111492911920779553925431482822

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