↑

Properties

Label 1-2925-2925.554-r0-0-0
Degree $1$
Conductor $2925$
Sign $-0.824 + 0.565i$
Analytic cond. $13.5836$
Root an. cond. $13.5836$
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  + (0.406 + 0.913i)2-s + (−0.669 + 0.743i)4-s + (−0.866 − 0.5i)7-s + (−0.951 − 0.309i)8-s + (0.406 + 0.913i)11-s + (0.104 − 0.994i)14-s + (−0.104 − 0.994i)16-s + (−0.309 + 0.951i)17-s + (0.951 + 0.309i)19-s + (−0.669 + 0.743i)22-s + (0.104 − 0.994i)23-s + (0.951 − 0.309i)28-s + (0.978 + 0.207i)29-s + (−0.207 − 0.978i)31-s + (0.866 − 0.5i)32-s + ⋯
L(s)  = 1  + (0.406 + 0.913i)2-s + (−0.669 + 0.743i)4-s + (−0.866 − 0.5i)7-s + (−0.951 − 0.309i)8-s + (0.406 + 0.913i)11-s + (0.104 − 0.994i)14-s + (−0.104 − 0.994i)16-s + (−0.309 + 0.951i)17-s + (0.951 + 0.309i)19-s + (−0.669 + 0.743i)22-s + (0.104 − 0.994i)23-s + (0.951 − 0.309i)28-s + (0.978 + 0.207i)29-s + (−0.207 − 0.978i)31-s + (0.866 − 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.824 + 0.565i$
Analytic conductor: \(13.5836\)
Root analytic conductor: \(13.5836\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (554, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2925,\ (0:\ ),\ -0.824 + 0.565i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4158411863 + 1.341499287i\)
\(L(\frac12)\) \(\approx\) \(0.4158411863 + 1.341499287i\)
\(L(1)\) \(\approx\) \(0.8784716229 + 0.6183508474i\)
\(L(1)\) \(\approx\) \(0.8784716229 + 0.6183508474i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.406 + 0.913i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.406 + 0.913i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (0.951 + 0.309i)T \)
23 \( 1 + (0.104 - 0.994i)T \)
29 \( 1 + (0.978 + 0.207i)T \)
31 \( 1 + (-0.207 - 0.978i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (0.406 - 0.913i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.207 + 0.978i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.406 + 0.913i)T \)
61 \( 1 + (0.913 - 0.406i)T \)
67 \( 1 + (0.207 + 0.978i)T \)
71 \( 1 + (0.951 - 0.309i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (-0.743 + 0.669i)T \)
89 \( 1 + (0.587 - 0.809i)T \)
97 \( 1 + (-0.207 + 0.978i)T \)
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

−18.94668271232455067170980881783, −18.353188405588430017813699988642, −17.70243115142431657014517169536, −16.69826495341168773409788190868, −15.81021000859058334740139764396, −15.45341671291745156111950949712, −14.29549255901095341261426098807, −13.768369531279301445056328347977, −13.22902378939196611541188107353, −12.38826254386542752939660162096, −11.61745454168443041709508671660, −11.35448033064373247006158287980, −10.18486008856499046653025729287, −9.68095082624185945160071901550, −8.959571568140642328797567738802, −8.354628495114597146817191883227, −6.98819642511936614297988335612, −6.37391455676821811568942859779, −5.400295724907303759083652851901, −4.97941656182618276672904069110, −3.69424686356049877120905925044, −3.225093753052571638774888585430, −2.54389469669331245940967417321, −1.421420220158585571473558922734, −0.463235101540658959317248105596, 0.95259549076994865777189789480, 2.33827458453601555830479653768, 3.31732250306116755218482824855, 4.07850948499889651376974988875, 4.62754784956870398987907978972, 5.71170377107945311469972183878, 6.35572031160027537595515858058, 7.02609562169123639870197468553, 7.627222834295719417541610011982, 8.52715782122362846268701471498, 9.32406807562223760290759088544, 9.93776279860639559170769333200, 10.76208366329172766940903613910, 11.93960023293073707238917176642, 12.57578859275254395628344495812, 13.02808974424786141051898342622, 13.9508320041253972722951190062, 14.452803964264931357727485770216, 15.25391239079541882843001284840, 15.90053252498177321217105003168, 16.527486994894588376084222245516, 17.19859977063666206209720328, 17.76188258096150948464356290031, 18.56263111616323657087528156836, 19.39265573855527010600334878193

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