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Properties

Label 2-380-76.31-c1-0-11
Degree $2$
Conductor $380$
Sign $0.992 - 0.124i$
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  + (0.270 − 1.38i)2-s + (0.435 + 0.755i)3-s + (−1.85 − 0.750i)4-s + (0.5 + 0.866i)5-s + (1.16 − 0.401i)6-s + 4.89i·7-s + (−1.54 + 2.37i)8-s + (1.11 − 1.93i)9-s + (1.33 − 0.460i)10-s + 0.00879i·11-s + (−0.241 − 1.72i)12-s + (0.420 + 0.242i)13-s + (6.80 + 1.32i)14-s + (−0.435 + 0.755i)15-s + (2.87 + 2.78i)16-s + (3.74 + 6.49i)17-s + ⋯
L(s)  = 1  + (0.191 − 0.981i)2-s + (0.251 + 0.435i)3-s + (−0.926 − 0.375i)4-s + (0.223 + 0.387i)5-s + (0.476 − 0.163i)6-s + 1.85i·7-s + (−0.545 + 0.838i)8-s + (0.373 − 0.646i)9-s + (0.422 − 0.145i)10-s + 0.00265i·11-s + (−0.0697 − 0.498i)12-s + (0.116 + 0.0673i)13-s + (1.81 + 0.353i)14-s + (−0.112 + 0.194i)15-s + (0.718 + 0.695i)16-s + (0.908 + 1.57i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.992 - 0.124i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.992 - 0.124i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51845 + 0.0946282i\)
\(L(\frac12)\) \(\approx\) \(1.51845 + 0.0946282i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.270 + 1.38i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-4.34 + 0.362i)T \)
good3 \( 1 + (-0.435 - 0.755i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 4.89iT - 7T^{2} \)
11 \( 1 - 0.00879iT - 11T^{2} \)
13 \( 1 + (-0.420 - 0.242i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.74 - 6.49i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (5.37 + 3.10i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.57 + 3.79i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.94T + 31T^{2} \)
37 \( 1 + 2.82iT - 37T^{2} \)
41 \( 1 + (-1.66 + 0.962i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.58 + 1.48i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.00 + 0.579i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.12 + 1.22i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.18 - 8.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.851 - 1.47i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.12 + 5.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.62 + 11.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.76 - 8.25i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.52 + 2.64i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.01iT - 83T^{2} \)
89 \( 1 + (4.11 + 2.37i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.51 - 3.76i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.51788547548015477834939172427, −10.34647068622281457370986388637, −9.652762433516965667913052746346, −8.954649213183462227552322002944, −8.055816288208735554214693695418, −6.12033251877754233285239254006, −5.55692424431071573452697485739, −4.07397079928322730753960028379, −3.07042632843012239430632852640, −1.90774740033938354062691504314, 1.06726018263050678333716672694, 3.43206721939862005607371297737, 4.56547650283652225485621400738, 5.48844318149304508939501420410, 6.94056296961860619329658406606, 7.50644392548210476461497501030, 8.069260931987069296798147637768, 9.568893763321525906810838464254, 10.05521106078292329084560068618, 11.39789286726532028937762514252

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