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Properties

Label 2-380-20.3-c1-0-31
Degree $2$
Conductor $380$
Sign $0.334 + 0.942i$
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  + (−1.01 − 0.979i)2-s + (0.582 − 0.582i)3-s + (0.0796 + 1.99i)4-s + (1.46 − 1.68i)5-s + (−1.16 + 0.0231i)6-s + (0.972 + 0.972i)7-s + (1.87 − 2.11i)8-s + 2.32i·9-s + (−3.14 + 0.282i)10-s − 4.73i·11-s + (1.21 + 1.11i)12-s + (4.00 + 4.00i)13-s + (−0.0387 − 1.94i)14-s + (−0.128 − 1.83i)15-s + (−3.98 + 0.318i)16-s + (1.22 − 1.22i)17-s + ⋯
L(s)  = 1  + (−0.721 − 0.692i)2-s + (0.336 − 0.336i)3-s + (0.0398 + 0.999i)4-s + (0.656 − 0.754i)5-s + (−0.475 + 0.00947i)6-s + (0.367 + 0.367i)7-s + (0.663 − 0.748i)8-s + 0.773i·9-s + (−0.995 + 0.0894i)10-s − 1.42i·11-s + (0.349 + 0.322i)12-s + (1.11 + 1.11i)13-s + (−0.0103 − 0.519i)14-s + (−0.0330 − 0.474i)15-s + (−0.996 + 0.0796i)16-s + (0.297 − 0.297i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.334 + 0.942i$
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: \((2,\ 380,\ (\ :1/2),\ 0.334 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03497 - 0.730711i\)
\(L(\frac12)\) \(\approx\) \(1.03497 - 0.730711i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.01 + 0.979i)T \)
5 \( 1 + (-1.46 + 1.68i)T \)
19 \( 1 - T \)
good3 \( 1 + (-0.582 + 0.582i)T - 3iT^{2} \)
7 \( 1 + (-0.972 - 0.972i)T + 7iT^{2} \)
11 \( 1 + 4.73iT - 11T^{2} \)
13 \( 1 + (-4.00 - 4.00i)T + 13iT^{2} \)
17 \( 1 + (-1.22 + 1.22i)T - 17iT^{2} \)
23 \( 1 + (1.83 - 1.83i)T - 23iT^{2} \)
29 \( 1 + 6.49iT - 29T^{2} \)
31 \( 1 + 1.38iT - 31T^{2} \)
37 \( 1 + (-3.47 + 3.47i)T - 37iT^{2} \)
41 \( 1 + 6.68T + 41T^{2} \)
43 \( 1 + (4.72 - 4.72i)T - 43iT^{2} \)
47 \( 1 + (6.12 + 6.12i)T + 47iT^{2} \)
53 \( 1 + (-3.65 - 3.65i)T + 53iT^{2} \)
59 \( 1 + 0.289T + 59T^{2} \)
61 \( 1 + 4.63T + 61T^{2} \)
67 \( 1 + (-3.20 - 3.20i)T + 67iT^{2} \)
71 \( 1 - 14.8iT - 71T^{2} \)
73 \( 1 + (-6.01 - 6.01i)T + 73iT^{2} \)
79 \( 1 - 5.24T + 79T^{2} \)
83 \( 1 + (4.42 - 4.42i)T - 83iT^{2} \)
89 \( 1 - 8.67iT - 89T^{2} \)
97 \( 1 + (3.14 - 3.14i)T - 97iT^{2} \)
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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

−11.35788196209074829408488106795, −10.18790839331619336594236554687, −9.210626243955752019174353082660, −8.472255019594340319131371989570, −7.990856633750625274392108267310, −6.48421043689531433181328434764, −5.29864341336983257004943547817, −3.86564086183553616081099600838, −2.38624693082565938651089413987, −1.29908570404430363771069266471, 1.57502541190854770093562582681, 3.30466970268517712474035314524, 4.83910486796133765354080914127, 6.05483019814456295461991020369, 6.84692107397443253037311502135, 7.81501797068082079965180614382, 8.786030846615817149995560599423, 9.802554062355743958901084140248, 10.27407324353499305495209317971, 11.07055716232248690371658617461

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