↑

Properties

Label 2-1805-5.4-c1-0-121
Degree $2$
Conductor $1805$
Sign $-0.319 + 0.947i$
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  − 0.471i·2-s − 1.04i·3-s + 1.77·4-s + (0.713 − 2.11i)5-s − 0.490·6-s + 1.17i·7-s − 1.78i·8-s + 1.91·9-s + (−1 − 0.336i)10-s + 0.713·11-s − 1.84i·12-s − 4.10i·13-s + 0.554·14-s + (−2.20 − 0.742i)15-s + 2.71·16-s + 2.55i·17-s + ⋯
L(s)  = 1  − 0.333i·2-s − 0.600i·3-s + 0.888·4-s + (0.319 − 0.947i)5-s − 0.200·6-s + 0.444i·7-s − 0.630i·8-s + 0.639·9-s + (−0.316 − 0.106i)10-s + 0.215·11-s − 0.533i·12-s − 1.13i·13-s + 0.148·14-s + (−0.569 − 0.191i)15-s + 0.678·16-s + 0.619i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.319 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.319 + 0.947i$
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: \((2,\ 1805,\ (\ :1/2),\ -0.319 + 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.531503430\)
\(L(\frac12)\) \(\approx\) \(2.531503430\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.713 + 2.11i)T \)
19 \( 1 \)
good2 \( 1 + 0.471iT - 2T^{2} \)
3 \( 1 + 1.04iT - 3T^{2} \)
7 \( 1 - 1.17iT - 7T^{2} \)
11 \( 1 - 0.713T + 11T^{2} \)
13 \( 1 + 4.10iT - 13T^{2} \)
17 \( 1 - 2.55iT - 17T^{2} \)
23 \( 1 - 0.607iT - 23T^{2} \)
29 \( 1 - 0.859T + 29T^{2} \)
31 \( 1 - 2.50T + 31T^{2} \)
37 \( 1 - 9.38iT - 37T^{2} \)
41 \( 1 + 4.12T + 41T^{2} \)
43 \( 1 + 10.1iT - 43T^{2} \)
47 \( 1 - 10.5iT - 47T^{2} \)
53 \( 1 + 4.98iT - 53T^{2} \)
59 \( 1 - 6.24T + 59T^{2} \)
61 \( 1 + 4.55T + 61T^{2} \)
67 \( 1 + 5.18iT - 67T^{2} \)
71 \( 1 + 13.1T + 71T^{2} \)
73 \( 1 - 12.3iT - 73T^{2} \)
79 \( 1 - 5.96T + 79T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 + 15.9T + 89T^{2} \)
97 \( 1 - 16.7iT - 97T^{2} \)
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

−8.969550490565918244813110290776, −8.169740090604501594878512164224, −7.53672188965176944791456484184, −6.56382692280126836927321879086, −5.94261537185870986302272970016, −5.05536622183067594480882547447, −3.91744215362673922122818733793, −2.76873810768300137109131327225, −1.77369981477163595671865182535, −0.983614181473396986055024569303, 1.61711170371885654179324319260, 2.62503434664485159513402950379, 3.66410933929767787615637388784, 4.52106541217634180960866776087, 5.62810289502732964472690498457, 6.53298366487694690750461540976, 7.06694597209201673107099416500, 7.57205305999646807032259838706, 8.814153357854531421194946568917, 9.682526135209119923264746971407

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