↑

Properties

Label 2-1890-63.25-c1-0-1
Degree $2$
Conductor $1890$
Sign $0.361 - 0.932i$
Analytic cond. $15.0917$
Root an. cond. $3.88480$
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-s + 4-s + (0.5 − 0.866i)5-s + (−2.25 − 1.38i)7-s − 8-s + (−0.5 + 0.866i)10-s + (0.480 + 0.832i)11-s + (−2.94 − 5.09i)13-s + (2.25 + 1.38i)14-s + 16-s + (−3.89 + 6.75i)17-s + (0.774 + 1.34i)19-s + (0.5 − 0.866i)20-s + (−0.480 − 0.832i)22-s + (−1.95 + 3.38i)23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (0.223 − 0.387i)5-s + (−0.853 − 0.521i)7-s − 0.353·8-s + (−0.158 + 0.273i)10-s + (0.144 + 0.251i)11-s + (−0.816 − 1.41i)13-s + (0.603 + 0.368i)14-s + 0.250·16-s + (−0.945 + 1.63i)17-s + (0.177 + 0.307i)19-s + (0.111 − 0.193i)20-s + (−0.102 − 0.177i)22-s + (−0.407 + 0.706i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.361 - 0.932i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.361 - 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6783219357\)
\(L(\frac12)\) \(\approx\) \(0.6783219357\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.25 + 1.38i)T \)
good11 \( 1 + (-0.480 - 0.832i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.94 + 5.09i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.89 - 6.75i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.774 - 1.34i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.95 - 3.38i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.543 + 0.940i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.55T + 31T^{2} \)
37 \( 1 + (-5.84 - 10.1i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.15 + 1.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.18 - 3.79i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.55T + 47T^{2} \)
53 \( 1 + (0.274 - 0.475i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.00T + 59T^{2} \)
61 \( 1 - 14.6T + 61T^{2} \)
67 \( 1 + 0.821T + 67T^{2} \)
71 \( 1 - 3.96T + 71T^{2} \)
73 \( 1 + (-0.368 + 0.638i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 6.79T + 79T^{2} \)
83 \( 1 + (6.12 - 10.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.85 - 4.94i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.80 + 13.5i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.596263786480147354648085594496, −8.453752779303221564556670724651, −8.014317078523237992651771443134, −7.06808787763423033553536329054, −6.29571177403039056241079779298, −5.55934420578204512858969401704, −4.37830714757928655462472452157, −3.39550680787265129553903511015, −2.32473178796400269658310387867, −1.03298514854004992516463744346, 0.35627101464286911363871909830, 2.23529017493026069773891204319, 2.68792140245558664447154062322, 4.02454354668073963995246167176, 5.09883354353982213469193479625, 6.16352052218469140101578549107, 6.88101284277884676969532617204, 7.25840426930450324060037622135, 8.552225109696593332785410695701, 9.232496041342899249844230323223

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