↑

Properties

Label 1-3549-3549.137-r0-0-0
Degree $1$
Conductor $3549$
Sign $0.659 - 0.751i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.663 + 0.748i)2-s + (−0.120 − 0.992i)4-s + (−0.721 − 0.692i)5-s + (0.822 + 0.568i)8-s + (0.996 − 0.0804i)10-s + (0.979 − 0.200i)11-s + (−0.970 + 0.239i)16-s + (0.568 − 0.822i)17-s + (0.866 − 0.5i)19-s + (−0.600 + 0.799i)20-s + (−0.5 + 0.866i)22-s + 23-s + (0.0402 + 0.999i)25-s + (0.200 − 0.979i)29-s + (0.534 − 0.845i)31-s + (0.464 − 0.885i)32-s + ⋯
L(s)  = 1  + (−0.663 + 0.748i)2-s + (−0.120 − 0.992i)4-s + (−0.721 − 0.692i)5-s + (0.822 + 0.568i)8-s + (0.996 − 0.0804i)10-s + (0.979 − 0.200i)11-s + (−0.970 + 0.239i)16-s + (0.568 − 0.822i)17-s + (0.866 − 0.5i)19-s + (−0.600 + 0.799i)20-s + (−0.5 + 0.866i)22-s + 23-s + (0.0402 + 0.999i)25-s + (0.200 − 0.979i)29-s + (0.534 − 0.845i)31-s + (0.464 − 0.885i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.659 - 0.751i$
Analytic conductor: \(16.4814\)
Root analytic conductor: \(16.4814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (0:\ ),\ 0.659 - 0.751i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.090398230 - 0.4937077080i\)
\(L(\frac12)\) \(\approx\) \(1.090398230 - 0.4937077080i\)
\(L(1)\) \(\approx\) \(0.8026482750 + 0.01127393917i\)
\(L(1)\) \(\approx\) \(0.8026482750 + 0.01127393917i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.663 + 0.748i)T \)
5 \( 1 + (-0.721 - 0.692i)T \)
11 \( 1 + (0.979 - 0.200i)T \)
17 \( 1 + (0.568 - 0.822i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + T \)
29 \( 1 + (0.200 - 0.979i)T \)
31 \( 1 + (0.534 - 0.845i)T \)
37 \( 1 + (-0.464 - 0.885i)T \)
41 \( 1 + (-0.160 - 0.987i)T \)
43 \( 1 + (0.845 - 0.534i)T \)
47 \( 1 + (-0.600 + 0.799i)T \)
53 \( 1 + (0.996 + 0.0804i)T \)
59 \( 1 + (0.239 - 0.970i)T \)
61 \( 1 + (0.428 - 0.903i)T \)
67 \( 1 + (-0.600 + 0.799i)T \)
71 \( 1 + (-0.774 + 0.632i)T \)
73 \( 1 + (0.316 - 0.948i)T \)
79 \( 1 + (0.799 + 0.600i)T \)
83 \( 1 + (-0.935 + 0.354i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.960 + 0.278i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.88682974739386972896237683972, −18.2263710911246935290735015734, −17.65990404761612365421724892793, −16.74004374297075421988042617210, −16.36250040048363526047774685553, −15.37401860863419872851636933894, −14.67597907632985221020145771734, −14.02932002313635677381314594074, −13.07774882119813953711115849104, −12.24665224797701571817546264071, −11.81229123408486989514419933814, −11.19139384507612941812740032836, −10.33871102632837406534529334616, −9.95908122878905334303423410750, −8.8912514965423077830831556697, −8.424338637863685041720482972570, −7.49343888550008197276682008395, −7.01140543665085631447720960049, −6.19969107501711781923998591301, −4.91892601435015070786707213477, −4.091412646308653896703255090112, −3.31623951939983624135605955456, −2.93184393921764763957788319701, −1.62394689856893066380911191540, −1.025788120898617991739913671269, 0.62931177195332342424911507761, 1.072047250934304634142121810016, 2.31361248733105540847648797500, 3.50915419908754035418383873778, 4.3441194893983282687334700611, 5.114972085231539707689559804805, 5.74331501258783220504922117387, 6.74782933257498496320353125459, 7.36830587412724475597404511847, 7.96096207163731007842410780350, 8.85994117625203426565211906679, 9.25664547447498668017529670881, 9.931916536469078718362191714177, 11.04154812193448620732272962533, 11.570932534714483068119698190425, 12.23213901531698178165591717654, 13.27575329523930336852854034694, 13.95015719010311745605327996508, 14.618055868467570530704843536736, 15.46952501467359569381467360303, 15.89991882179664745657069266818, 16.58600616174004074185515547448, 17.20113124307318944724412110135, 17.70502148504450871468685323919, 18.78668716232152511465350571245

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