Properties

Label 2-380-380.159-c2-0-47
Degree $2$
Conductor $380$
Sign $0.996 - 0.0795i$
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  + (−1.42 + 1.40i)2-s + (1.00 + 1.74i)3-s + (0.0597 − 3.99i)4-s + (−4.76 + 1.51i)5-s + (−3.89 − 1.07i)6-s − 3.07·7-s + (5.52 + 5.78i)8-s + (2.46 − 4.26i)9-s + (4.66 − 8.84i)10-s − 3.77i·11-s + (7.05 − 3.93i)12-s + (−9.11 − 5.26i)13-s + (4.38 − 4.31i)14-s + (−7.46 − 6.80i)15-s + (−15.9 − 0.477i)16-s + (9.93 − 5.73i)17-s + ⋯
L(s)  = 1  + (−0.712 + 0.701i)2-s + (0.336 + 0.582i)3-s + (0.0149 − 0.999i)4-s + (−0.952 + 0.303i)5-s + (−0.648 − 0.179i)6-s − 0.439·7-s + (0.691 + 0.722i)8-s + (0.273 − 0.473i)9-s + (0.466 − 0.884i)10-s − 0.342i·11-s + (0.587 − 0.327i)12-s + (−0.701 − 0.404i)13-s + (0.312 − 0.308i)14-s + (−0.497 − 0.453i)15-s + (−0.999 − 0.0298i)16-s + (0.584 − 0.337i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0795i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.996 - 0.0795i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ 0.996 - 0.0795i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.902940 + 0.0359695i\)
\(L(\frac12)\) \(\approx\) \(0.902940 + 0.0359695i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.42 - 1.40i)T \)
5 \( 1 + (4.76 - 1.51i)T \)
19 \( 1 + (-16.4 - 9.51i)T \)
good3 \( 1 + (-1.00 - 1.74i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + 3.07T + 49T^{2} \)
11 \( 1 + 3.77iT - 121T^{2} \)
13 \( 1 + (9.11 + 5.26i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-9.93 + 5.73i)T + (144.5 - 250. i)T^{2} \)
23 \( 1 + (-6.96 + 12.0i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (9.09 - 15.7i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + 50.1iT - 961T^{2} \)
37 \( 1 + 17.0iT - 1.36e3T^{2} \)
41 \( 1 + (-35.7 - 61.9i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-2.56 - 4.44i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-27.8 + 48.1i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-21.0 - 12.1i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-79.5 + 45.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (2.51 - 4.36i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-16.9 + 29.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-112. + 65.1i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (69.2 - 39.9i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-6.25 + 3.61i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 61.2T + 6.88e3T^{2} \)
89 \( 1 + (-14.2 + 24.6i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-19.9 + 11.5i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.90837170893430148386024361737, −9.903529984563944242772548430655, −9.429659686543938721480227725269, −8.280133321934760370230725304070, −7.52621412765925260039667363589, −6.64522824860485525140573763882, −5.40987714503872090116722331772, −4.15296856398048951723945249903, −2.97419478226393231083618713410, −0.59731838164428550604633282326, 1.13963637380455320481080765990, 2.59343081180506312560525820077, 3.77242668975886401287744458731, 4.98067656307610951149255051000, 7.05511079102119836545443265775, 7.43098106458665796853274610077, 8.366481074894310491070251132371, 9.274603326507266491571750496393, 10.17765566454886219531958568813, 11.17478922648117572428958184826

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