↑

Properties

Label 2-385-7.2-c1-0-0
Degree $2$
Conductor $385$
Sign $-0.835 - 0.550i$
Analytic cond. $3.07424$
Root an. cond. $1.75335$
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  + (−0.533 + 0.923i)2-s + (−1.53 − 2.66i)3-s + (0.431 + 0.747i)4-s + (0.5 − 0.866i)5-s + 3.27·6-s + (−2.62 − 0.323i)7-s − 3.05·8-s + (−3.22 + 5.59i)9-s + (0.533 + 0.923i)10-s + (−0.5 − 0.866i)11-s + (1.32 − 2.29i)12-s + 2.06·13-s + (1.69 − 2.25i)14-s − 3.07·15-s + (0.763 − 1.32i)16-s + (0.561 + 0.972i)17-s + ⋯
L(s)  = 1  + (−0.376 + 0.652i)2-s + (−0.887 − 1.53i)3-s + (0.215 + 0.373i)4-s + (0.223 − 0.387i)5-s + 1.33·6-s + (−0.992 − 0.122i)7-s − 1.07·8-s + (−1.07 + 1.86i)9-s + (0.168 + 0.291i)10-s + (−0.150 − 0.261i)11-s + (0.383 − 0.663i)12-s + 0.571·13-s + (0.453 − 0.601i)14-s − 0.793·15-s + (0.190 − 0.330i)16-s + (0.136 + 0.235i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $-0.835 - 0.550i$
Analytic conductor: \(3.07424\)
Root analytic conductor: \(1.75335\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 385,\ (\ :1/2),\ -0.835 - 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0422641 + 0.140947i\)
\(L(\frac12)\) \(\approx\) \(0.0422641 + 0.140947i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.62 + 0.323i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.533 - 0.923i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.53 + 2.66i)T + (-1.5 + 2.59i)T^{2} \)
13 \( 1 - 2.06T + 13T^{2} \)
17 \( 1 + (-0.561 - 0.972i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.23 - 7.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.99 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.01T + 29T^{2} \)
31 \( 1 + (-0.162 - 0.281i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.70 - 4.68i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.23T + 41T^{2} \)
43 \( 1 + 12.0T + 43T^{2} \)
47 \( 1 + (-0.526 + 0.912i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.442 - 0.765i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.93 + 12.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.819 - 1.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.37 + 9.31i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.18T + 71T^{2} \)
73 \( 1 + (-6.48 - 11.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.71 - 6.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.20T + 83T^{2} \)
89 \( 1 + (-7.04 + 12.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 16.8T + 97T^{2} \)
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

−11.99278224009936525629244566776, −11.00009883122546570220499439870, −9.804990800717534055764990113285, −8.447936990440236576062494092053, −7.910072253564413995195062904813, −6.85654830782597819358918130641, −6.21166172472947859555506204489, −5.63224402390362294920831129527, −3.48340111503240800510011616259, −1.77378727745682749579165528130, 0.11477521458045915360295248156, 2.64340495865992989756903299970, 3.78508534768212304222268710648, 5.07386342165838432452927650041, 6.08595462822434162245481107101, 6.72007213909022362348971596595, 8.928262121253223511649661553263, 9.412805650180621861682024667091, 10.32208548440610884514507239458, 10.69364923103972052249520250717

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