Properties

Label 2-380-20.19-c2-0-97
Degree $2$
Conductor $380$
Sign $0.322 + 0.946i$
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

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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 + (−9.98 − 0.493i)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 + (−0.998 − 0.0493i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.322 + 0.946i$
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: \((2,\ 380,\ (\ :1),\ 0.322 + 0.946i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.88972 - 2.06921i\)
\(L(\frac12)\) \(\approx\) \(2.88972 - 2.06921i\)
\(L(2)\) 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 \( 1 - 94.6iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.07089533609267113930377593777, −10.50142401462086714011138959253, −8.886462484835445692779873943335, −8.038387559287082460094255534745, −7.49095755710559416753568276036, −5.69041458592142149682150397075, −4.92761384093920863482325779037, −3.85216917103821012014316083382, −2.77799190133153759678858310353, −1.25144119502280895901823158187, 2.07082842437161307730824791497, 3.27863821130354740036402435044, 4.41302411015534824032187799207, 5.16493606923527756002914592307, 6.77839793205642571339821267124, 7.47874457883171130347446394507, 8.360562950360869386876119344946, 9.058164893012224045133989647720, 11.08335076635103508851591108203, 11.39288199125667923780923106573

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