Properties

Label 2-380-380.303-c2-0-65
Degree $2$
Conductor $380$
Sign $-0.895 - 0.445i$
Analytic cond. $10.3542$
Root an. cond. $3.21780$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.912 + 1.77i)2-s + (3.66 + 3.66i)3-s + (−2.33 + 3.24i)4-s + (4.37 + 2.42i)5-s + (−3.18 + 9.87i)6-s + (2.51 + 2.51i)7-s + (−7.91 − 1.18i)8-s + 17.9i·9-s + (−0.322 + 9.99i)10-s − 14.1i·11-s + (−20.4 + 3.35i)12-s + (0.192 − 0.192i)13-s + (−2.18 + 6.77i)14-s + (7.14 + 24.9i)15-s + (−5.10 − 15.1i)16-s + (18.2 − 18.2i)17-s + ⋯
L(s)  = 1  + (0.456 + 0.889i)2-s + (1.22 + 1.22i)3-s + (−0.583 + 0.812i)4-s + (0.874 + 0.484i)5-s + (−0.530 + 1.64i)6-s + (0.359 + 0.359i)7-s + (−0.988 − 0.148i)8-s + 1.99i·9-s + (−0.0322 + 0.999i)10-s − 1.28i·11-s + (−1.70 + 0.279i)12-s + (0.0148 − 0.0148i)13-s + (−0.155 + 0.484i)14-s + (0.476 + 1.66i)15-s + (−0.319 − 0.947i)16-s + (1.07 − 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.895 - 0.445i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ -0.895 - 0.445i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.790571 + 3.36419i\)
\(L(\frac12)\) \(\approx\) \(0.790571 + 3.36419i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.912 - 1.77i)T \)
5 \( 1 + (-4.37 - 2.42i)T \)
19 \( 1 + (17.8 + 6.50i)T \)
good3 \( 1 + (-3.66 - 3.66i)T + 9iT^{2} \)
7 \( 1 + (-2.51 - 2.51i)T + 49iT^{2} \)
11 \( 1 + 14.1iT - 121T^{2} \)
13 \( 1 + (-0.192 + 0.192i)T - 169iT^{2} \)
17 \( 1 + (-18.2 + 18.2i)T - 289iT^{2} \)
23 \( 1 + (-18.5 + 18.5i)T - 529iT^{2} \)
29 \( 1 - 31.6T + 841T^{2} \)
31 \( 1 + 35.7T + 961T^{2} \)
37 \( 1 + (17.7 + 17.7i)T + 1.36e3iT^{2} \)
41 \( 1 - 49.7iT - 1.68e3T^{2} \)
43 \( 1 + (13.5 - 13.5i)T - 1.84e3iT^{2} \)
47 \( 1 + (37.4 + 37.4i)T + 2.20e3iT^{2} \)
53 \( 1 + (20.7 - 20.7i)T - 2.80e3iT^{2} \)
59 \( 1 + 17.7iT - 3.48e3T^{2} \)
61 \( 1 + 85.5T + 3.72e3T^{2} \)
67 \( 1 + (28.1 - 28.1i)T - 4.48e3iT^{2} \)
71 \( 1 - 62.1T + 5.04e3T^{2} \)
73 \( 1 + (24.6 + 24.6i)T + 5.32e3iT^{2} \)
79 \( 1 - 116. iT - 6.24e3T^{2} \)
83 \( 1 + (26.7 - 26.7i)T - 6.88e3iT^{2} \)
89 \( 1 + 149.T + 7.92e3T^{2} \)
97 \( 1 + (-9.47 - 9.47i)T + 9.40e3iT^{2} \)
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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

−11.34337870585770422292255305649, −10.36927024335465148158164143563, −9.396823326160298961983569581762, −8.755097922868400914693628319780, −8.054128364144586013886872677949, −6.73511680703477050246307006122, −5.53361929785770169943379416328, −4.74614364215030076899593500479, −3.36407319004921064244440146531, −2.71325215185556836208929218779, 1.43350465096777843073162599401, 1.90033117905014090967421929275, 3.23850005389075595523433982902, 4.55194349794121360594236950027, 5.87483157250187241187873326453, 7.00580204849768985515649562603, 8.118596386784554013921817890769, 8.971942509510779635754405334634, 9.804423727726683869408800535645, 10.65651346786366227893491131166

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