↑

Properties

Label 2-1805-5.4-c1-0-32
Degree $2$
Conductor $1805$
Sign $1$
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  − 2.26i·2-s + 3.11i·3-s − 3.13·4-s − 2.23·5-s + 7.05·6-s + 2.57i·8-s − 6.67·9-s + 5.06i·10-s + 5.95·11-s − 9.75i·12-s − 2.42i·13-s − 6.95i·15-s − 0.435·16-s + 15.1i·18-s + 7.01·20-s + ⋯
L(s)  = 1  − 1.60i·2-s + 1.79i·3-s − 1.56·4-s − 0.999·5-s + 2.87·6-s + 0.910i·8-s − 2.22·9-s + 1.60i·10-s + 1.79·11-s − 2.81i·12-s − 0.672i·13-s − 1.79i·15-s − 0.108·16-s + 3.56i·18-s + 1.56·20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $1$
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: \((2,\ 1805,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.225321874\)
\(L(\frac12)\) \(\approx\) \(1.225321874\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 2.23T \)
19 \( 1 \)
good2 \( 1 + 2.26iT - 2T^{2} \)
3 \( 1 - 3.11iT - 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 5.95T + 11T^{2} \)
13 \( 1 + 2.42iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 0.802iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 13.2iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 15.5T + 61T^{2} \)
67 \( 1 - 16.0iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 19.4iT - 97T^{2} \)
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

−9.406636663983155385334254624264, −8.982343093507879259116952712917, −8.261269119261341358688662367025, −6.88335113609748911542064933028, −5.59373270242351146340662772633, −4.55359827550319081931796536084, −4.00982241277590427094699627361, −3.55598562744537075653650478976, −2.68882304434766406854133344512, −0.930290577331421066070869268059, 0.63767766363901653730748255257, 1.94039344724753856668005228458, 3.55738768756499499497728199727, 4.54091445309630917871547875185, 5.70557207497996675281629429474, 6.52441692402346116355613254201, 6.91047647404518929253915286127, 7.39268273993563867985294883215, 8.296643445542158284293999788016, 8.652331982526005982709777555926

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