↑

Properties

Label 1-1859-1859.63-r0-0-0
Degree $1$
Conductor $1859$
Sign $0.305 - 0.952i$
Analytic cond. $8.63315$
Root an. cond. $8.63315$
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  + (−0.889 − 0.457i)2-s + (−0.877 − 0.478i)3-s + (0.581 + 0.813i)4-s + (−0.421 − 0.906i)5-s + (0.561 + 0.827i)6-s + (−0.832 − 0.554i)7-s + (−0.144 − 0.989i)8-s + (0.541 + 0.840i)9-s + (−0.0402 + 0.999i)10-s + (−0.120 − 0.992i)12-s + (0.485 + 0.873i)14-s + (−0.0643 + 0.997i)15-s + (−0.324 + 0.945i)16-s + (0.937 − 0.347i)17-s + (−0.0965 − 0.995i)18-s + (0.406 − 0.913i)19-s + ⋯
L(s)  = 1  + (−0.889 − 0.457i)2-s + (−0.877 − 0.478i)3-s + (0.581 + 0.813i)4-s + (−0.421 − 0.906i)5-s + (0.561 + 0.827i)6-s + (−0.832 − 0.554i)7-s + (−0.144 − 0.989i)8-s + (0.541 + 0.840i)9-s + (−0.0402 + 0.999i)10-s + (−0.120 − 0.992i)12-s + (0.485 + 0.873i)14-s + (−0.0643 + 0.997i)15-s + (−0.324 + 0.945i)16-s + (0.937 − 0.347i)17-s + (−0.0965 − 0.995i)18-s + (0.406 − 0.913i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $0.305 - 0.952i$
Analytic conductor: \(8.63315\)
Root analytic conductor: \(8.63315\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1859,\ (0:\ ),\ 0.305 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4738916398 - 0.3458267095i\)
\(L(\frac12)\) \(\approx\) \(0.4738916398 - 0.3458267095i\)
\(L(1)\) \(\approx\) \(0.4435822882 - 0.2432450138i\)
\(L(1)\) \(\approx\) \(0.4435822882 - 0.2432450138i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.889 - 0.457i)T \)
3 \( 1 + (-0.877 - 0.478i)T \)
5 \( 1 + (-0.421 - 0.906i)T \)
7 \( 1 + (-0.832 - 0.554i)T \)
17 \( 1 + (0.937 - 0.347i)T \)
19 \( 1 + (0.406 - 0.913i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.704 + 0.709i)T \)
31 \( 1 + (-0.849 - 0.527i)T \)
37 \( 1 + (0.176 + 0.984i)T \)
41 \( 1 + (-0.478 + 0.877i)T \)
43 \( 1 + (0.692 + 0.721i)T \)
47 \( 1 + (0.0965 - 0.995i)T \)
53 \( 1 + (0.715 - 0.698i)T \)
59 \( 1 + (0.0161 + 0.999i)T \)
61 \( 1 + (-0.247 + 0.968i)T \)
67 \( 1 + (0.316 + 0.948i)T \)
71 \( 1 + (0.972 - 0.231i)T \)
73 \( 1 + (0.548 + 0.836i)T \)
79 \( 1 + (0.861 - 0.506i)T \)
83 \( 1 + (0.794 + 0.607i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.996 + 0.0884i)T \)
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

−20.048838137608512043750857243429, −19.126080090301049103453041366240, −18.800754650507167524112159388075, −18.06230485483921820379459954188, −17.329165077057925846519616885805, −16.57152019650250946354739092345, −15.86943865273232786437007115827, −15.465725625591048386684467639798, −14.706757718717186804342030853767, −13.91359510461356533506555721961, −12.36630960216817853785747552952, −12.03960561982748423703741617430, −10.98682402856887214550075272636, −10.54909279869611247166344258503, −9.71241785357481015919332373189, −9.25465370923962071987411644686, −8.028460994845546578819660408986, −7.28200993489139690995410335167, −6.52900805381020469531078073042, −5.84191459643290440435497340114, −5.28756181587830341269720450038, −3.82032989891189109845479161590, −3.137869700746499742999865190856, −1.8936799558873060991779413824, −0.55709489044436612793842574473, 0.75205611994566024097768919505, 1.10038385570412697057472076607, 2.487748789464281696922456803431, 3.485055754365953223980491175238, 4.47932187460552112866734727816, 5.353693903390499865888224485731, 6.491229604549062986589862219454, 7.11769297886371779992963089411, 7.80838652984581252084336938500, 8.67725274795033899631541897613, 9.556308583788069034006978292265, 10.21594072615586563139533022488, 11.04960199601423483250967644847, 11.77308295047176725039663186354, 12.39123965362375192840305670491, 13.02219056173986677754182742826, 13.588616425349016191880880174968, 15.13496426901558055838413484764, 16.146676821524648995876544721658, 16.50635844863487661236274959274, 16.9093418676825724531515962944, 17.83465103164597253140302778373, 18.49868671556102414048168241836, 19.25582910556141015461783614502, 19.83351822091234716090484307118

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