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Properties

Label 2-380-20.7-c1-0-43
Degree $2$
Conductor $380$
Sign $0.420 - 0.907i$
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  + (1.25 + 0.647i)2-s + (1.78 + 1.78i)3-s + (1.16 + 1.62i)4-s + (0.0829 − 2.23i)5-s + (1.09 + 3.40i)6-s + (0.904 − 0.904i)7-s + (0.408 + 2.79i)8-s + 3.39i·9-s + (1.55 − 2.75i)10-s − 4.92i·11-s + (−0.831 + 4.98i)12-s + (−4.64 + 4.64i)13-s + (1.72 − 0.551i)14-s + (4.14 − 3.84i)15-s + (−1.29 + 3.78i)16-s + (−3.50 − 3.50i)17-s + ⋯
L(s)  = 1  + (0.889 + 0.457i)2-s + (1.03 + 1.03i)3-s + (0.581 + 0.813i)4-s + (0.0370 − 0.999i)5-s + (0.445 + 1.39i)6-s + (0.341 − 0.341i)7-s + (0.144 + 0.989i)8-s + 1.13i·9-s + (0.490 − 0.871i)10-s − 1.48i·11-s + (−0.240 + 1.44i)12-s + (−1.28 + 1.28i)13-s + (0.460 − 0.147i)14-s + (1.07 − 0.993i)15-s + (−0.324 + 0.945i)16-s + (−0.850 − 0.850i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.420 - 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.420 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.420 - 0.907i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.420 - 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.47798 + 1.58260i\)
\(L(\frac12)\) \(\approx\) \(2.47798 + 1.58260i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.25 - 0.647i)T \)
5 \( 1 + (-0.0829 + 2.23i)T \)
19 \( 1 - T \)
good3 \( 1 + (-1.78 - 1.78i)T + 3iT^{2} \)
7 \( 1 + (-0.904 + 0.904i)T - 7iT^{2} \)
11 \( 1 + 4.92iT - 11T^{2} \)
13 \( 1 + (4.64 - 4.64i)T - 13iT^{2} \)
17 \( 1 + (3.50 + 3.50i)T + 17iT^{2} \)
23 \( 1 + (2.65 + 2.65i)T + 23iT^{2} \)
29 \( 1 - 4.23iT - 29T^{2} \)
31 \( 1 - 6.52iT - 31T^{2} \)
37 \( 1 + (4.68 + 4.68i)T + 37iT^{2} \)
41 \( 1 - 3.99T + 41T^{2} \)
43 \( 1 + (-1.41 - 1.41i)T + 43iT^{2} \)
47 \( 1 + (-0.801 + 0.801i)T - 47iT^{2} \)
53 \( 1 + (2.72 - 2.72i)T - 53iT^{2} \)
59 \( 1 - 13.9T + 59T^{2} \)
61 \( 1 - 6.61T + 61T^{2} \)
67 \( 1 + (-1.41 + 1.41i)T - 67iT^{2} \)
71 \( 1 + 0.666iT - 71T^{2} \)
73 \( 1 + (-9.09 + 9.09i)T - 73iT^{2} \)
79 \( 1 - 7.09T + 79T^{2} \)
83 \( 1 + (-6.54 - 6.54i)T + 83iT^{2} \)
89 \( 1 + 6.08iT - 89T^{2} \)
97 \( 1 + (-1.52 - 1.52i)T + 97iT^{2} \)
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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

−11.64768912966034002752778045782, −10.68119288884330581719662283030, −9.300934762496144283179004566413, −8.816430030899330076042584439705, −7.940644481257237363791850912712, −6.76295761666078895331892143240, −5.24680529908922767464086316646, −4.56075507482889084243121782119, −3.71326608253944766158731042833, −2.41775521445474733404515759143, 2.10805820574957766206879981832, 2.47969895074218418789817568139, 3.85894722943346207855740056834, 5.28552689053865197340199134356, 6.57063899434174529858586780944, 7.36258965145802676142361049764, 8.029232020635166245150135279063, 9.669074446756987020559377760391, 10.24690650640115397803984266374, 11.47436826777095062107008959533

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