↑

Properties

Label 2-1805-95.29-c0-0-2
Degree $2$
Conductor $1805$
Sign $-0.934 + 0.356i$
Analytic cond. $0.900812$
Root an. cond. $0.949111$
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  + (−0.245 − 1.39i)2-s + (1.08 − 0.909i)3-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)5-s + (−1.53 − 1.28i)6-s + (0.173 − 0.984i)9-s + (0.245 − 1.39i)10-s + (−0.707 + 1.22i)12-s + (−1.08 − 0.909i)13-s + (1.32 − 0.483i)15-s + (−0.766 + 0.642i)16-s − 1.41·18-s − 1.00·20-s + (0.766 + 0.642i)25-s + (−1 + 1.73i)26-s + ⋯
L(s)  = 1  + (−0.245 − 1.39i)2-s + (1.08 − 0.909i)3-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)5-s + (−1.53 − 1.28i)6-s + (0.173 − 0.984i)9-s + (0.245 − 1.39i)10-s + (−0.707 + 1.22i)12-s + (−1.08 − 0.909i)13-s + (1.32 − 0.483i)15-s + (−0.766 + 0.642i)16-s − 1.41·18-s − 1.00·20-s + (0.766 + 0.642i)25-s + (−1 + 1.73i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.934 + 0.356i$
Analytic conductor: \(0.900812\)
Root analytic conductor: \(0.949111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (694, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :0),\ -0.934 + 0.356i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.562096559\)
\(L(\frac12)\) \(\approx\) \(1.562096559\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 \)
good2 \( 1 + (0.245 + 1.39i)T + (-0.939 + 0.342i)T^{2} \)
3 \( 1 + (-1.08 + 0.909i)T + (0.173 - 0.984i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (1.08 + 0.909i)T + (0.173 + 0.984i)T^{2} \)
17 \( 1 + (0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - 1.41T + T^{2} \)
41 \( 1 + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (1.32 - 0.483i)T + (0.766 - 0.642i)T^{2} \)
59 \( 1 + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.245 - 1.39i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (-0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.173 - 0.984i)T^{2} \)
97 \( 1 + (-0.245 - 1.39i)T + (-0.939 + 0.342i)T^{2} \)
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

−9.362990682295778217072194223741, −8.520601719086243962885986919751, −7.69362194174688517125657967119, −6.92497573980798380719760080275, −6.01019068878331903172543146300, −4.82476533243821939300251494868, −3.44862906504784738210147429331, −2.69445650531879961926806446593, −2.23112060488153625285272238018, −1.20985613021069030088407408016, 2.08485033141097938306299473611, 2.99236678600978075624871626404, 4.44355090341399305087240742690, 4.87849137352910908635234844209, 5.93528344901466757414730812043, 6.64804393676525250117807880003, 7.58750761423763056242872720851, 8.272155889277577999804461200559, 9.062485401713371965755796871273, 9.485915738674189445202001009979

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