Properties

Label 2-1805-5.4-c1-0-93
Degree $2$
Conductor $1805$
Sign $-0.929 - 0.369i$
Analytic cond. $14.4129$
Root an. cond. $3.79644$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.41i·2-s − 0.537i·3-s − 3.85·4-s + (−2.07 − 0.826i)5-s − 1.29·6-s + 3.18i·7-s + 4.49i·8-s + 2.71·9-s + (−2 + 5.02i)10-s + 4.15·11-s + 2.07i·12-s − 2.07i·13-s + 7.71·14-s + (−0.443 + 1.11i)15-s + 3.15·16-s − 5.79i·17-s + ⋯
L(s)  = 1  − 1.71i·2-s − 0.310i·3-s − 1.92·4-s + (−0.929 − 0.369i)5-s − 0.530·6-s + 1.20i·7-s + 1.58i·8-s + 0.903·9-s + (−0.632 + 1.58i)10-s + 1.25·11-s + 0.597i·12-s − 0.574i·13-s + 2.06·14-s + (−0.114 + 0.288i)15-s + 0.788·16-s − 1.40i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.929 - 0.369i$
Analytic conductor: \(14.4129\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (1084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ -0.929 - 0.369i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.287027449\)
\(L(\frac12)\) \(\approx\) \(1.287027449\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.07 + 0.826i)T \)
19 \( 1 \)
good2 \( 1 + 2.41iT - 2T^{2} \)
3 \( 1 + 0.537iT - 3T^{2} \)
7 \( 1 - 3.18iT - 7T^{2} \)
11 \( 1 - 4.15T + 11T^{2} \)
13 \( 1 + 2.07iT - 13T^{2} \)
17 \( 1 + 5.79iT - 17T^{2} \)
23 \( 1 + 2.60iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 2.59T + 31T^{2} \)
37 \( 1 - 4.30iT - 37T^{2} \)
41 \( 1 - 0.599T + 41T^{2} \)
43 \( 1 - 3.18iT - 43T^{2} \)
47 \( 1 + 11.7iT - 47T^{2} \)
53 \( 1 + 11.7iT - 53T^{2} \)
59 \( 1 + 1.71T + 59T^{2} \)
61 \( 1 + 8.75T + 61T^{2} \)
67 \( 1 - 4.76iT - 67T^{2} \)
71 \( 1 + 13.7T + 71T^{2} \)
73 \( 1 + 2.72iT - 73T^{2} \)
79 \( 1 + 1.40T + 79T^{2} \)
83 \( 1 + 7.07iT - 83T^{2} \)
89 \( 1 - 16.5T + 89T^{2} \)
97 \( 1 - 2.07iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.947155999282684307226545114000, −8.501320771828631491737294669643, −7.38383122863973894201064014406, −6.47157302246313282062509013641, −5.08129568667083050599157896054, −4.48504821491565818703401380171, −3.53230361095427239104790116697, −2.72044275827066815376355793963, −1.61099698428263238427734397148, −0.58369764332420719095883884932, 1.23957675572701137288802427525, 3.64631606416388633308605903292, 4.19036253732449386584103905424, 4.61693235331069114780833830615, 6.11066428776698731758569280567, 6.62924133858571111476306348373, 7.40460647885912661252227599668, 7.70349621175330964154329825677, 8.766938755025454326618737491434, 9.363023213588967663552981567946

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