↑

Properties

Label 1-1805-1805.144-r0-0-0
Degree $1$
Conductor $1805$
Sign $-0.315 + 0.949i$
Analytic cond. $8.38238$
Root an. cond. $8.38238$
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  + (−0.451 + 0.892i)2-s + (0.754 + 0.656i)3-s + (−0.592 − 0.805i)4-s + (−0.926 + 0.376i)6-s + (0.401 + 0.915i)7-s + (0.986 − 0.164i)8-s + (0.137 + 0.990i)9-s + (0.789 + 0.614i)11-s + (0.0825 − 0.996i)12-s + (0.191 − 0.981i)13-s + (−0.998 − 0.0550i)14-s + (−0.298 + 0.954i)16-s + (0.592 − 0.805i)17-s + (−0.945 − 0.324i)18-s + (−0.298 + 0.954i)21-s + (−0.904 + 0.426i)22-s + ⋯
L(s)  = 1  + (−0.451 + 0.892i)2-s + (0.754 + 0.656i)3-s + (−0.592 − 0.805i)4-s + (−0.926 + 0.376i)6-s + (0.401 + 0.915i)7-s + (0.986 − 0.164i)8-s + (0.137 + 0.990i)9-s + (0.789 + 0.614i)11-s + (0.0825 − 0.996i)12-s + (0.191 − 0.981i)13-s + (−0.998 − 0.0550i)14-s + (−0.298 + 0.954i)16-s + (0.592 − 0.805i)17-s + (−0.945 − 0.324i)18-s + (−0.298 + 0.954i)21-s + (−0.904 + 0.426i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.315 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.315 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.315 + 0.949i$
Analytic conductor: \(8.38238\)
Root analytic conductor: \(8.38238\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1805,\ (0:\ ),\ -0.315 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.145494756 + 1.587264593i\)
\(L(\frac12)\) \(\approx\) \(1.145494756 + 1.587264593i\)
\(L(1)\) \(\approx\) \(0.9858239033 + 0.7611733390i\)
\(L(1)\) \(\approx\) \(0.9858239033 + 0.7611733390i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.451 + 0.892i)T \)
3 \( 1 + (0.754 + 0.656i)T \)
7 \( 1 + (0.401 + 0.915i)T \)
11 \( 1 + (0.789 + 0.614i)T \)
13 \( 1 + (0.191 - 0.981i)T \)
17 \( 1 + (0.592 - 0.805i)T \)
23 \( 1 + (0.754 - 0.656i)T \)
29 \( 1 + (0.993 + 0.110i)T \)
31 \( 1 + (0.546 - 0.837i)T \)
37 \( 1 + (-0.789 - 0.614i)T \)
41 \( 1 + (0.0275 + 0.999i)T \)
43 \( 1 + (-0.716 + 0.697i)T \)
47 \( 1 + (0.926 - 0.376i)T \)
53 \( 1 + (0.926 - 0.376i)T \)
59 \( 1 + (0.0275 + 0.999i)T \)
61 \( 1 + (0.635 - 0.771i)T \)
67 \( 1 + (-0.350 + 0.936i)T \)
71 \( 1 + (0.635 + 0.771i)T \)
73 \( 1 + (0.592 - 0.805i)T \)
79 \( 1 + (0.716 - 0.697i)T \)
83 \( 1 + (0.677 - 0.735i)T \)
89 \( 1 + (-0.592 - 0.805i)T \)
97 \( 1 + (-0.350 - 0.936i)T \)
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

−19.71956150976509233326251719265, −19.335873872822591006125799274182, −18.8224242965989146960815063394, −17.87827222213448284611104276730, −17.14319632672489864725266480423, −16.72942958971189929500685835143, −15.49026875832991256753148701213, −14.320755810013316322208102527445, −13.89296149126233105508511756698, −13.42369794907829333073215015726, −12.30329448180899057515298325090, −11.86966383155272734961993115698, −10.94875364931786085215845621012, −10.21597427866011502861198249501, −9.28752272074208929944681482154, −8.630671065310430018565817531437, −8.04839116608425717016642126293, −7.100983491228306451050103906121, −6.52970061120786547910797807855, −5.01851503244841294414831722409, −3.774133135280177764234958044383, −3.66468968891966167066001883146, −2.43066774829052779659221633601, −1.40384540210264942806475783169, −1.019645860049266028549704666532, 1.05384281468258537131787221960, 2.223725860750707419111558153706, 3.14864175548873949658000613649, 4.3382885889585607616094799578, 5.004494959426379926438424152418, 5.71256373871807716441139632469, 6.770366059059836336801869648658, 7.648182338281986195895273379356, 8.38800504381224884423523396577, 8.91469805132526266661337133068, 9.70253831697174139125578763569, 10.26333314318913968245550141496, 11.25128951714462599178173663819, 12.2676953298273828737314296931, 13.2550337017844978984647259239, 14.08576410746144313340668593314, 14.83249366580335108614258967354, 15.115562153182076696685110718226, 15.9061700217036984515394689240, 16.58092933928425680795624287348, 17.451619194609459102665215426064, 18.15844377334378884136210303771, 18.86925952081501752156990802965, 19.59927591391811198265987907203, 20.33407822053384967750122967834

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