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Properties

Label 2-380-76.27-c1-0-25
Degree $2$
Conductor $380$
Sign $0.0436 + 0.999i$
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  + (−1.38 − 0.278i)2-s + (−0.418 + 0.725i)3-s + (1.84 + 0.773i)4-s + (0.5 − 0.866i)5-s + (0.782 − 0.888i)6-s − 3.40i·7-s + (−2.34 − 1.58i)8-s + (1.14 + 1.99i)9-s + (−0.934 + 1.06i)10-s − 4.46i·11-s + (−1.33 + 1.01i)12-s + (−5.48 + 3.16i)13-s + (−0.950 + 4.72i)14-s + (0.418 + 0.725i)15-s + (2.80 + 2.85i)16-s + (1.89 − 3.27i)17-s + ⋯
L(s)  = 1  + (−0.980 − 0.197i)2-s + (−0.241 + 0.418i)3-s + (0.922 + 0.386i)4-s + (0.223 − 0.387i)5-s + (0.319 − 0.362i)6-s − 1.28i·7-s + (−0.827 − 0.560i)8-s + (0.383 + 0.663i)9-s + (−0.295 + 0.335i)10-s − 1.34i·11-s + (−0.384 + 0.292i)12-s + (−1.52 + 0.879i)13-s + (−0.254 + 1.26i)14-s + (0.108 + 0.187i)15-s + (0.700 + 0.713i)16-s + (0.458 − 0.794i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0436 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0436 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.0436 + 0.999i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.0436 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.502422 - 0.480967i\)
\(L(\frac12)\) \(\approx\) \(0.502422 - 0.480967i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 + 0.278i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (3.32 + 2.82i)T \)
good3 \( 1 + (0.418 - 0.725i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 3.40iT - 7T^{2} \)
11 \( 1 + 4.46iT - 11T^{2} \)
13 \( 1 + (5.48 - 3.16i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.89 + 3.27i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-4.26 + 2.46i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.61 + 2.08i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.00T + 31T^{2} \)
37 \( 1 + 10.4iT - 37T^{2} \)
41 \( 1 + (0.135 + 0.0782i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.00 + 1.73i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.57 - 2.06i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.15 + 5.28i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.36 + 2.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.39 - 2.41i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.90 - 6.76i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.95 + 10.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.48 - 7.76i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.02 - 8.69i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.68iT - 83T^{2} \)
89 \( 1 + (5.46 - 3.15i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.13 - 4.11i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.92084598211797490478827447537, −10.20633219501495780419235251224, −9.471391775347723688653087623243, −8.479359933291840003127359620613, −7.41184177655744188450618293910, −6.74039153231460152446781637721, −5.18179785264257308047595007809, −4.08151948044494313989439290178, −2.47192212214743595276956115093, −0.65029223957181436253154380155, 1.74562160297306956961399140245, 2.87850280293823126649322825838, 5.07044195593039815283302801597, 6.11815606588478217498082952539, 6.95361254598489117820825357376, 7.80223264091747770552238015375, 8.850539112101189705325992102528, 9.949384445317436221169192960304, 10.17008873308691383688229272406, 11.72256572308940932919733736252

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