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Properties

Label 2-3549-273.212-c0-0-2
Degree $2$
Conductor $3549$
Sign $-0.313 - 0.949i$
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  + (0.5 + 0.866i)2-s + i·3-s + (0.5 − 0.866i)5-s + (−0.866 + 0.5i)6-s + i·7-s + 8-s − 9-s + 0.999·10-s + (−0.866 + 0.5i)14-s + (0.866 + 0.5i)15-s + (0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (−0.5 − 0.866i)18-s − 21-s + (−0.866 + 0.5i)23-s + i·24-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + i·3-s + (0.5 − 0.866i)5-s + (−0.866 + 0.5i)6-s + i·7-s + 8-s − 9-s + 0.999·10-s + (−0.866 + 0.5i)14-s + (0.866 + 0.5i)15-s + (0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (−0.5 − 0.866i)18-s − 21-s + (−0.866 + 0.5i)23-s + i·24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $-0.313 - 0.949i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ -0.313 - 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.015641736\)
\(L(\frac12)\) \(\approx\) \(2.015641736\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 - iT \)
13 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + T^{2} \)
17 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
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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

−9.103092940020306275793268120299, −8.088487723884077032696419746803, −7.69036473185514031166210394920, −6.15053182024886870340032740231, −5.93345847437284757693237798966, −5.32837057828373150430931037778, −4.62223249983119886887002533969, −3.89511367400376132366025915412, −2.66094251729593760371351664440, −1.53640221788056927017860448308, 1.11105433886405313539100166715, 2.08532218667761347037789871417, 2.89077439677812238058584500060, 3.50957579461671003904981039525, 4.52010504140580781161881329044, 5.51498301649279536582998431953, 6.53644818081287854729792018963, 6.92051921130517510715475489611, 7.77358289121868493209114027457, 8.154025428344701014300445314741

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