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Properties

Label 1-3549-3549.38-r0-0-0
Degree $1$
Conductor $3549$
Sign $-0.958 + 0.284i$
Analytic cond. $16.4814$
Root an. cond. $16.4814$
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  + (−0.0402 + 0.999i)2-s + (−0.996 − 0.0804i)4-s + (0.632 + 0.774i)5-s + (0.120 − 0.992i)8-s + (−0.799 + 0.600i)10-s + (−0.0402 − 0.999i)11-s + (0.987 + 0.160i)16-s + (−0.919 + 0.391i)17-s + (−0.5 + 0.866i)19-s + (−0.568 − 0.822i)20-s + 22-s + (0.5 − 0.866i)23-s + (−0.200 + 0.979i)25-s + (−0.885 + 0.464i)29-s + (0.948 + 0.316i)31-s + (−0.200 + 0.979i)32-s + ⋯
L(s)  = 1  + (−0.0402 + 0.999i)2-s + (−0.996 − 0.0804i)4-s + (0.632 + 0.774i)5-s + (0.120 − 0.992i)8-s + (−0.799 + 0.600i)10-s + (−0.0402 − 0.999i)11-s + (0.987 + 0.160i)16-s + (−0.919 + 0.391i)17-s + (−0.5 + 0.866i)19-s + (−0.568 − 0.822i)20-s + 22-s + (0.5 − 0.866i)23-s + (−0.200 + 0.979i)25-s + (−0.885 + 0.464i)29-s + (0.948 + 0.316i)31-s + (−0.200 + 0.979i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $-0.958 + 0.284i$
Analytic conductor: \(16.4814\)
Root analytic conductor: \(16.4814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (0:\ ),\ -0.958 + 0.284i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1783232277 + 1.228008441i\)
\(L(\frac12)\) \(\approx\) \(0.1783232277 + 1.228008441i\)
\(L(1)\) \(\approx\) \(0.7653454358 + 0.5950426608i\)
\(L(1)\) \(\approx\) \(0.7653454358 + 0.5950426608i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.0402 + 0.999i)T \)
5 \( 1 + (0.632 + 0.774i)T \)
11 \( 1 + (-0.0402 - 0.999i)T \)
17 \( 1 + (-0.919 + 0.391i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.885 + 0.464i)T \)
31 \( 1 + (0.948 + 0.316i)T \)
37 \( 1 + (0.200 + 0.979i)T \)
41 \( 1 + (0.970 - 0.239i)T \)
43 \( 1 + (-0.748 - 0.663i)T \)
47 \( 1 + (0.996 - 0.0804i)T \)
53 \( 1 + (0.919 - 0.391i)T \)
59 \( 1 + (-0.987 + 0.160i)T \)
61 \( 1 + (0.919 + 0.391i)T \)
67 \( 1 + (-0.428 + 0.903i)T \)
71 \( 1 + (-0.970 + 0.239i)T \)
73 \( 1 + (-0.0402 - 0.999i)T \)
79 \( 1 + (-0.996 + 0.0804i)T \)
83 \( 1 + (0.970 + 0.239i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.354 - 0.935i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.36453112846641936737211137915, −17.60651569186437491201113910787, −17.43185232808605587553496987493, −16.59233018142935468490506438419, −15.55832935770368535902044553316, −14.9354707359548522055764456459, −13.96114471156313834840035924208, −13.29235064204873516618393377141, −12.94870654318718647118026585168, −12.18915300648771692810732808028, −11.443618582241325798301126476179, −10.78574792512537603086104609259, −9.90883500585135094225692703179, −9.31190957685700760811942477859, −8.95081379030158502589220652293, −7.97109345427552989092095282161, −7.14055372943832783834511689295, −6.06761816378633053427882997141, −5.245008291951420568342238618841, −4.53925030242094798469785217357, −4.086220731861373495395809917392, −2.765612264618821578272094722248, −2.201441993989559291566026701292, −1.44286039586242945867808687522, −0.41311138349278364747032591797, 1.00754369749647356101466594896, 2.15732622462072198522034640217, 3.13222233602912662899614494671, 3.9223184397500903843612351341, 4.800397363860603583380420004805, 5.778209995789976610131125621204, 6.168284683215331484512512490209, 6.85813641967567869040033185472, 7.57932847764404338441417430322, 8.64281464533419086331741037383, 8.81695948716021657919442106091, 9.986261427203474044308644939997, 10.474720714657428334446895347710, 11.18404731906033179128529933879, 12.26040416475570379657139813722, 13.22955789444904048170263662609, 13.53694718269270215493520337613, 14.39762097166272089027362395728, 14.84438142371894917163662492411, 15.52427913469899295943644314920, 16.39395937308822781241872231606, 16.93389656834778177333214885485, 17.55850814254535695226970817338, 18.30200372175656546640714592581, 18.86464207933871680041293186593

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