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Properties

Label 2-380-20.3-c1-0-32
Degree $2$
Conductor $380$
Sign $0.268 + 0.963i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.615 − 1.27i)2-s + (1.93 − 1.93i)3-s + (−1.24 + 1.56i)4-s + (−0.168 + 2.22i)5-s + (−3.66 − 1.27i)6-s + (2.91 + 2.91i)7-s + (2.76 + 0.615i)8-s − 4.51i·9-s + (2.94 − 1.15i)10-s − 3.32i·11-s + (0.632 + 5.44i)12-s + (−0.313 − 0.313i)13-s + (1.91 − 5.51i)14-s + (3.99 + 4.64i)15-s + (−0.916 − 3.89i)16-s + (3.64 − 3.64i)17-s + ⋯
L(s)  = 1  + (−0.435 − 0.900i)2-s + (1.11 − 1.11i)3-s + (−0.620 + 0.783i)4-s + (−0.0753 + 0.997i)5-s + (−1.49 − 0.520i)6-s + (1.10 + 1.10i)7-s + (0.976 + 0.217i)8-s − 1.50i·9-s + (0.930 − 0.366i)10-s − 1.00i·11-s + (0.182 + 1.57i)12-s + (−0.0870 − 0.0870i)13-s + (0.512 − 1.47i)14-s + (1.03 + 1.20i)15-s + (−0.229 − 0.973i)16-s + (0.885 − 0.885i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.268 + 0.963i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.268 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29052 - 0.979836i\)
\(L(\frac12)\) \(\approx\) \(1.29052 - 0.979836i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.615 + 1.27i)T \)
5 \( 1 + (0.168 - 2.22i)T \)
19 \( 1 + T \)
good3 \( 1 + (-1.93 + 1.93i)T - 3iT^{2} \)
7 \( 1 + (-2.91 - 2.91i)T + 7iT^{2} \)
11 \( 1 + 3.32iT - 11T^{2} \)
13 \( 1 + (0.313 + 0.313i)T + 13iT^{2} \)
17 \( 1 + (-3.64 + 3.64i)T - 17iT^{2} \)
23 \( 1 + (-2.24 + 2.24i)T - 23iT^{2} \)
29 \( 1 - 8.34iT - 29T^{2} \)
31 \( 1 + 0.286iT - 31T^{2} \)
37 \( 1 + (6.20 - 6.20i)T - 37iT^{2} \)
41 \( 1 - 4.08T + 41T^{2} \)
43 \( 1 + (1.71 - 1.71i)T - 43iT^{2} \)
47 \( 1 + (8.21 + 8.21i)T + 47iT^{2} \)
53 \( 1 + (-0.195 - 0.195i)T + 53iT^{2} \)
59 \( 1 + 1.88T + 59T^{2} \)
61 \( 1 - 6.98T + 61T^{2} \)
67 \( 1 + (-0.844 - 0.844i)T + 67iT^{2} \)
71 \( 1 + 6.69iT - 71T^{2} \)
73 \( 1 + (6.74 + 6.74i)T + 73iT^{2} \)
79 \( 1 + 16.9T + 79T^{2} \)
83 \( 1 + (4.69 - 4.69i)T - 83iT^{2} \)
89 \( 1 + 9.12iT - 89T^{2} \)
97 \( 1 + (5.27 - 5.27i)T - 97iT^{2} \)
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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

−11.37108144803201815503856783163, −10.33353834103941725746160686257, −9.043660748340323859986565328974, −8.440737920213486236802403863244, −7.77127605309097382302876213576, −6.82022589980584551668385037068, −5.22883575341831077921192795480, −3.28618969734590633731233623082, −2.69348584927177274174335130904, −1.55364792067490420100454202344, 1.59663929193629028776125144845, 4.06081871019706159805751269012, 4.45527948181902134420674870043, 5.47711680522056582823184056007, 7.31595237153858042174626424908, 8.026256806736599200087545770342, 8.618275260691670949277497039407, 9.685391055452337366926819843159, 10.08412642374670522288829499184, 11.15568893494360812033466373398

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