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Properties

Label 2-380-19.4-c1-0-4
Degree $2$
Conductor $380$
Sign $-0.129 + 0.991i$
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  + (−0.275 − 1.56i)3-s + (0.766 − 0.642i)5-s + (0.778 − 1.34i)7-s + (0.454 − 0.165i)9-s + (−1.44 − 2.50i)11-s + (−0.501 + 2.84i)13-s + (−1.21 − 1.01i)15-s + (−3.43 − 1.25i)17-s + (3.58 − 2.48i)19-s + (−2.32 − 0.845i)21-s + (−1.02 − 0.860i)23-s + (0.173 − 0.984i)25-s + (−2.76 − 4.78i)27-s + (4.25 − 1.54i)29-s + (−0.0994 + 0.172i)31-s + ⋯
L(s)  = 1  + (−0.159 − 0.902i)3-s + (0.342 − 0.287i)5-s + (0.294 − 0.509i)7-s + (0.151 − 0.0550i)9-s + (−0.435 − 0.754i)11-s + (−0.139 + 0.788i)13-s + (−0.313 − 0.263i)15-s + (−0.834 − 0.303i)17-s + (0.821 − 0.570i)19-s + (−0.506 − 0.184i)21-s + (−0.213 − 0.179i)23-s + (0.0347 − 0.196i)25-s + (−0.531 − 0.920i)27-s + (0.789 − 0.287i)29-s + (−0.0178 + 0.0309i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.129 + 0.991i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.129 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.881518 - 1.00453i\)
\(L(\frac12)\) \(\approx\) \(0.881518 - 1.00453i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (-3.58 + 2.48i)T \)
good3 \( 1 + (0.275 + 1.56i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (-0.778 + 1.34i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.44 + 2.50i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.501 - 2.84i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (3.43 + 1.25i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (1.02 + 0.860i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-4.25 + 1.54i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (0.0994 - 0.172i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.14T + 37T^{2} \)
41 \( 1 + (-0.240 - 1.36i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-1.02 + 0.861i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (1.00 - 0.366i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-5.96 - 5.00i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-4.57 - 1.66i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-7.26 - 6.09i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-7.36 + 2.68i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-1.21 + 1.02i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-2.01 - 11.4i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-2.38 - 13.5i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-5.84 + 10.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.54 - 14.4i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (0.990 + 0.360i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.27223878268465338033190225327, −10.24500181812451582149622605705, −9.200319219498462402936378653157, −8.237158493992482828423537922633, −7.20034983584266748588889962068, −6.53517948109928316944572769069, −5.31889912830912896785960148951, −4.15389739339759859109569725829, −2.41650122820220261354097770447, −0.973920049023747323381824895310, 2.10338935061628216512316786486, 3.56683132656891231639246382393, 4.85922325379728790549941884514, 5.51894443152191514538584294636, 6.86367009834831307519157020446, 7.937063924529617612693000975656, 9.021935525783366853009398075949, 10.05510355106884000045819882252, 10.39493294690944895461755582469, 11.45586570745066665466083604605

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