Properties

Label 2-380-20.19-c2-0-27
Degree $2$
Conductor $380$
Sign $0.925 + 0.379i$
Analytic cond. $10.3542$
Root an. cond. $3.21780$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.85 + 0.736i)2-s − 2.04·3-s + (2.91 − 2.73i)4-s + (−4.55 + 2.06i)5-s + (3.80 − 1.50i)6-s − 11.4·7-s + (−3.39 + 7.24i)8-s − 4.82·9-s + (6.93 − 7.20i)10-s + 2.74i·11-s + (−5.95 + 5.60i)12-s + 23.9i·13-s + (21.2 − 8.43i)14-s + (9.30 − 4.22i)15-s + (0.985 − 15.9i)16-s − 18.6i·17-s + ⋯
L(s)  = 1  + (−0.929 + 0.368i)2-s − 0.681·3-s + (0.728 − 0.684i)4-s + (−0.910 + 0.413i)5-s + (0.633 − 0.251i)6-s − 1.63·7-s + (−0.424 + 0.905i)8-s − 0.535·9-s + (0.693 − 0.720i)10-s + 0.249i·11-s + (−0.496 + 0.466i)12-s + 1.83i·13-s + (1.52 − 0.602i)14-s + (0.620 − 0.281i)15-s + (0.0615 − 0.998i)16-s − 1.09i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.925 + 0.379i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ 0.925 + 0.379i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.186982 - 0.0368888i\)
\(L(\frac12)\) \(\approx\) \(0.186982 - 0.0368888i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.85 - 0.736i)T \)
5 \( 1 + (4.55 - 2.06i)T \)
19 \( 1 - 4.35iT \)
good3 \( 1 + 2.04T + 9T^{2} \)
7 \( 1 + 11.4T + 49T^{2} \)
11 \( 1 - 2.74iT - 121T^{2} \)
13 \( 1 - 23.9iT - 169T^{2} \)
17 \( 1 + 18.6iT - 289T^{2} \)
23 \( 1 + 33.3T + 529T^{2} \)
29 \( 1 + 14.3T + 841T^{2} \)
31 \( 1 + 6.39iT - 961T^{2} \)
37 \( 1 + 9.68iT - 1.36e3T^{2} \)
41 \( 1 - 8.81T + 1.68e3T^{2} \)
43 \( 1 - 32.1T + 1.84e3T^{2} \)
47 \( 1 + 11.9T + 2.20e3T^{2} \)
53 \( 1 + 101. iT - 2.80e3T^{2} \)
59 \( 1 - 51.3iT - 3.48e3T^{2} \)
61 \( 1 - 41.5T + 3.72e3T^{2} \)
67 \( 1 - 91.4T + 4.48e3T^{2} \)
71 \( 1 - 93.7iT - 5.04e3T^{2} \)
73 \( 1 - 77.6iT - 5.32e3T^{2} \)
79 \( 1 - 59.7iT - 6.24e3T^{2} \)
83 \( 1 + 53.7T + 6.88e3T^{2} \)
89 \( 1 - 16.6T + 7.92e3T^{2} \)
97 \( 1 + 94.6iT - 9.40e3T^{2} \)
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

−11.18007086385706749776863142273, −9.963106239242941428384003591073, −9.396488072978737690735399602440, −8.326627732362602713848985622139, −7.00458646394901768133282319899, −6.68681474493668661567182487911, −5.64351892163678801370978135155, −4.00394685047830651800215819770, −2.55048931850673154599663100807, −0.23781611321743118075385563085, 0.58922380475769216991470482754, 2.94786726414379337484448453758, 3.78345627955884141384150486461, 5.69518078979594100021484025028, 6.42841920338916825682942539042, 7.72274775496118802202663459950, 8.401370146847631464635721962459, 9.440706357726736092791071472655, 10.43844711028149141523990303399, 10.95702021178720923227905216181

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