Properties

Label 2-380-380.59-c1-0-47
Degree $2$
Conductor $380$
Sign $-0.761 + 0.648i$
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  + (−1.36 − 0.369i)2-s + (1.57 − 1.87i)3-s + (1.72 + 1.00i)4-s + (0.572 − 2.16i)5-s + (−2.83 + 1.97i)6-s + (−0.614 − 1.06i)7-s + (−1.98 − 2.01i)8-s + (−0.516 − 2.92i)9-s + (−1.58 + 2.73i)10-s + (−4.39 − 2.53i)11-s + (4.60 − 1.64i)12-s + (−2.56 + 2.15i)13-s + (0.445 + 1.67i)14-s + (−3.14 − 4.46i)15-s + (1.96 + 3.48i)16-s + (2.68 + 0.472i)17-s + ⋯
L(s)  = 1  + (−0.965 − 0.261i)2-s + (0.907 − 1.08i)3-s + (0.863 + 0.504i)4-s + (0.256 − 0.966i)5-s + (−1.15 + 0.806i)6-s + (−0.232 − 0.402i)7-s + (−0.701 − 0.712i)8-s + (−0.172 − 0.976i)9-s + (−0.499 + 0.866i)10-s + (−1.32 − 0.765i)11-s + (1.32 − 0.475i)12-s + (−0.711 + 0.597i)13-s + (0.118 + 0.448i)14-s + (−0.812 − 1.15i)15-s + (0.490 + 0.871i)16-s + (0.650 + 0.114i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.761 + 0.648i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.761 + 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.370084 - 1.00547i\)
\(L(\frac12)\) \(\approx\) \(0.370084 - 1.00547i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.369i)T \)
5 \( 1 + (-0.572 + 2.16i)T \)
19 \( 1 + (-4.32 + 0.522i)T \)
good3 \( 1 + (-1.57 + 1.87i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (0.614 + 1.06i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.39 + 2.53i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.56 - 2.15i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-2.68 - 0.472i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (-1.81 - 0.661i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (1.46 - 0.258i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.22 - 3.85i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.08T + 37T^{2} \)
41 \( 1 + (-6.26 + 7.47i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (6.67 - 2.42i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.254 + 1.44i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (6.76 + 2.46i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-0.802 + 4.55i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (9.63 + 3.50i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (1.39 - 0.246i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-12.4 + 4.51i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-2.85 + 3.40i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (-9.39 - 7.88i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-8.42 - 14.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.86 - 2.22i)T + (-15.4 + 87.6i)T^{2} \)
97 \( 1 + (-0.425 + 2.41i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.89781962281607510106752173092, −9.777572511264299083074794917013, −9.103894659147978005291039286460, −8.038964307363461104034810008563, −7.74386290765796284097586964688, −6.65291057519030479839013981898, −5.25274200129080871530503736747, −3.30007872855165442659516012212, −2.20376966573430182990323359611, −0.874562879738710722593713860946, 2.53575865068991994044271679892, 3.06762279458027426133040045941, 4.95048214374728448626307096448, 6.04119883669502040938939437356, 7.52178103518425910560188523161, 7.88501323213504938859192197604, 9.287079515897843468783381245381, 9.851090753193691911265730706603, 10.28322937608569193790649152277, 11.23433277067673588437051457039

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