↑

Properties

Label 2-1840-5.4-c1-0-19
Degree $2$
Conductor $1840$
Sign $-0.234 - 0.972i$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
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  + 2.80i·3-s + (−2.17 + 0.523i)5-s − 4.50i·7-s − 4.84·9-s + 4.10·11-s + 4.10i·13-s + (−1.46 − 6.09i)15-s − 2.26i·17-s + 6.77·19-s + 12.6·21-s + i·23-s + (4.45 − 2.27i)25-s − 5.17i·27-s − 4.13·29-s − 1.84·31-s + ⋯
L(s)  = 1  + 1.61i·3-s + (−0.972 + 0.234i)5-s − 1.70i·7-s − 1.61·9-s + 1.23·11-s + 1.13i·13-s + (−0.378 − 1.57i)15-s − 0.549i·17-s + 1.55·19-s + 2.75·21-s + 0.208i·23-s + (0.890 − 0.455i)25-s − 0.996i·27-s − 0.768·29-s − 0.330·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.234 - 0.972i$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ -0.234 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.424235829\)
\(L(\frac12)\) \(\approx\) \(1.424235829\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.17 - 0.523i)T \)
23 \( 1 - iT \)
good3 \( 1 - 2.80iT - 3T^{2} \)
7 \( 1 + 4.50iT - 7T^{2} \)
11 \( 1 - 4.10T + 11T^{2} \)
13 \( 1 - 4.10iT - 13T^{2} \)
17 \( 1 + 2.26iT - 17T^{2} \)
19 \( 1 - 6.77T + 19T^{2} \)
29 \( 1 + 4.13T + 29T^{2} \)
31 \( 1 + 1.84T + 31T^{2} \)
37 \( 1 - 11.1iT - 37T^{2} \)
41 \( 1 - 8.36T + 41T^{2} \)
43 \( 1 - 5.43iT - 43T^{2} \)
47 \( 1 - 0.593iT - 47T^{2} \)
53 \( 1 + 1.70iT - 53T^{2} \)
59 \( 1 - 6.19T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 - 5.78iT - 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 - 0.363iT - 73T^{2} \)
79 \( 1 - 1.75T + 79T^{2} \)
83 \( 1 + 9.72iT - 83T^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 - 4.38iT - 97T^{2} \)
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

−9.520119019359327302780203188599, −9.033167277149879098852687819193, −7.82234214666369430498086902715, −7.19223277506642326108201092844, −6.41039492447069727796802477445, −4.96066369496754285291750263373, −4.36481169274196277309409654208, −3.77555603973991431669182301722, −3.24373507635631722451011804905, −1.06658553904558180934472198866, 0.67270569309599845651983317144, 1.81847951678729403683005820594, 2.85153720424181199580208034890, 3.79356696820118665198037532380, 5.36346632080729829015145317533, 5.81531310398971229098165896736, 6.73383261907973668254917879321, 7.61377132290990997090246815392, 8.011764176904122516875564663994, 8.938803023979623554906654728801

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