↑

Properties

Label 2-380-95.18-c1-0-1
Degree $2$
Conductor $380$
Sign $-0.877 - 0.480i$
Analytic cond. $3.03431$
Root an. cond. $1.74192$
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  + (−1.49 + 1.49i)3-s + (0.311 + 2.21i)5-s + (0.311 − 0.311i)7-s − 1.47i·9-s + 2.90·11-s + (−2.84 + 2.84i)13-s + (−3.77 − 2.84i)15-s + (−2.52 + 2.52i)17-s + (−4.34 − 0.377i)19-s + 0.930i·21-s + (−4.11 − 4.11i)23-s + (−4.80 + 1.37i)25-s + (−2.28 − 2.28i)27-s + 2.99·29-s + 0.930i·31-s + ⋯
L(s)  = 1  + (−0.863 + 0.863i)3-s + (0.139 + 0.990i)5-s + (0.117 − 0.117i)7-s − 0.491i·9-s + 0.875·11-s + (−0.789 + 0.789i)13-s + (−0.975 − 0.735i)15-s + (−0.612 + 0.612i)17-s + (−0.996 − 0.0866i)19-s + 0.203i·21-s + (−0.858 − 0.858i)23-s + (−0.961 + 0.275i)25-s + (−0.439 − 0.439i)27-s + 0.555·29-s + 0.167i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 - 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.877 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.877 - 0.480i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.877 - 0.480i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.195907 + 0.766006i\)
\(L(\frac12)\) \(\approx\) \(0.195907 + 0.766006i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.311 - 2.21i)T \)
19 \( 1 + (4.34 + 0.377i)T \)
good3 \( 1 + (1.49 - 1.49i)T - 3iT^{2} \)
7 \( 1 + (-0.311 + 0.311i)T - 7iT^{2} \)
11 \( 1 - 2.90T + 11T^{2} \)
13 \( 1 + (2.84 - 2.84i)T - 13iT^{2} \)
17 \( 1 + (2.52 - 2.52i)T - 17iT^{2} \)
23 \( 1 + (4.11 + 4.11i)T + 23iT^{2} \)
29 \( 1 - 2.99T + 29T^{2} \)
31 \( 1 - 0.930iT - 31T^{2} \)
37 \( 1 + (-8.11 - 8.11i)T + 37iT^{2} \)
41 \( 1 + 2.06iT - 41T^{2} \)
43 \( 1 + (-2.11 - 2.11i)T + 43iT^{2} \)
47 \( 1 + (2.73 - 2.73i)T - 47iT^{2} \)
53 \( 1 + (0.565 - 0.565i)T - 53iT^{2} \)
59 \( 1 - 9.32T + 59T^{2} \)
61 \( 1 - 3.52T + 61T^{2} \)
67 \( 1 + (0.144 + 0.144i)T + 67iT^{2} \)
71 \( 1 + 9.61iT - 71T^{2} \)
73 \( 1 + (4.09 + 4.09i)T + 73iT^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 + (-9.21 - 9.21i)T + 83iT^{2} \)
89 \( 1 + 7.55T + 89T^{2} \)
97 \( 1 + (-12.0 - 12.0i)T + 97iT^{2} \)
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

−11.49941089041308122370950965213, −10.80062302686562414791773060185, −10.13028845345308979183069840888, −9.301447981425570116910412920510, −8.002735003982296323166655737661, −6.60258025326314724804305744377, −6.21639895938220611241593208454, −4.68668510363022814883233852627, −4.01152806929620290341627454987, −2.28951635460406794908975101408, 0.57630209287723318488852689996, 2.04549200203088946772299984299, 4.12194454440904079765700438179, 5.29442651367902507928002538904, 6.09331500614647913395511120059, 7.09253586672282545499486229239, 8.099528329534150845597514636747, 9.122244400538696569160584078076, 10.01462975982780887925965039379, 11.36642439373054006443118913559

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