Properties

Label 2-380-20.19-c2-0-67
Degree $2$
Conductor $380$
Sign $0.997 + 0.0719i$
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  + (−1.94 + 0.485i)2-s + 4.64·3-s + (3.52 − 1.88i)4-s + (2.66 − 4.22i)5-s + (−9.01 + 2.25i)6-s + 9.90·7-s + (−5.92 + 5.37i)8-s + 12.6·9-s + (−3.12 + 9.50i)10-s − 2.36i·11-s + (16.4 − 8.76i)12-s + 16.5i·13-s + (−19.2 + 4.81i)14-s + (12.4 − 19.6i)15-s + (8.89 − 13.3i)16-s + 21.4i·17-s + ⋯
L(s)  = 1  + (−0.970 + 0.242i)2-s + 1.54·3-s + (0.881 − 0.471i)4-s + (0.533 − 0.845i)5-s + (−1.50 + 0.376i)6-s + 1.41·7-s + (−0.741 + 0.671i)8-s + 1.40·9-s + (−0.312 + 0.950i)10-s − 0.214i·11-s + (1.36 − 0.730i)12-s + 1.27i·13-s + (−1.37 + 0.343i)14-s + (0.826 − 1.31i)15-s + (0.555 − 0.831i)16-s + 1.26i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0719i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.997 + 0.0719i$
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: \((2,\ 380,\ (\ :1),\ 0.997 + 0.0719i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.32282 - 0.0837008i\)
\(L(\frac12)\) \(\approx\) \(2.32282 - 0.0837008i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.94 - 0.485i)T \)
5 \( 1 + (-2.66 + 4.22i)T \)
19 \( 1 + 4.35iT \)
good3 \( 1 - 4.64T + 9T^{2} \)
7 \( 1 - 9.90T + 49T^{2} \)
11 \( 1 + 2.36iT - 121T^{2} \)
13 \( 1 - 16.5iT - 169T^{2} \)
17 \( 1 - 21.4iT - 289T^{2} \)
23 \( 1 - 6.86T + 529T^{2} \)
29 \( 1 + 36.1T + 841T^{2} \)
31 \( 1 + 46.4iT - 961T^{2} \)
37 \( 1 - 17.7iT - 1.36e3T^{2} \)
41 \( 1 + 54.6T + 1.68e3T^{2} \)
43 \( 1 - 4.27T + 1.84e3T^{2} \)
47 \( 1 + 40.9T + 2.20e3T^{2} \)
53 \( 1 + 10.7iT - 2.80e3T^{2} \)
59 \( 1 + 70.4iT - 3.48e3T^{2} \)
61 \( 1 - 101.T + 3.72e3T^{2} \)
67 \( 1 - 7.29T + 4.48e3T^{2} \)
71 \( 1 - 31.8iT - 5.04e3T^{2} \)
73 \( 1 + 83.7iT - 5.32e3T^{2} \)
79 \( 1 - 137. iT - 6.24e3T^{2} \)
83 \( 1 - 99.8T + 6.88e3T^{2} \)
89 \( 1 - 76.9T + 7.92e3T^{2} \)
97 \( 1 - 155. iT - 9.40e3T^{2} \)
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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

−10.96730693446928064330713855385, −9.695550693322557973824216639156, −9.166842981831612152226446681642, −8.210089851582431965465470597433, −8.091421724168947227806638758018, −6.70119857520699316189703016242, −5.30207165172616259202167556146, −3.99767067821051581936771585202, −2.12672637265642202552526702456, −1.58531980122600837341571291867, 1.60434570369741559471588226590, 2.59722784185836365605253194567, 3.46387609273912708004953869389, 5.31070881479901246325638523607, 7.03239436644726168111619824548, 7.66136732825883664454413791312, 8.397786324478029188601063527538, 9.237816591289480078569122095219, 10.10323771283043687808812157221, 10.84918044158843177630418508906

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