Properties

Label 2-380-19.5-c1-0-3
Degree $2$
Conductor $380$
Sign $0.773 + 0.633i$
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  + (−0.0781 + 0.443i)3-s + (−0.766 − 0.642i)5-s + (−1.06 − 1.85i)7-s + (2.62 + 0.956i)9-s + (2.25 − 3.90i)11-s + (−0.155 − 0.881i)13-s + (0.344 − 0.289i)15-s + (−0.00854 + 0.00311i)17-s + (4.31 − 0.640i)19-s + (0.904 − 0.329i)21-s + (4.94 − 4.15i)23-s + (0.173 + 0.984i)25-s + (−1.30 + 2.25i)27-s + (−0.381 − 0.138i)29-s + (−0.920 − 1.59i)31-s + ⋯
L(s)  = 1  + (−0.0451 + 0.255i)3-s + (−0.342 − 0.287i)5-s + (−0.404 − 0.700i)7-s + (0.876 + 0.318i)9-s + (0.680 − 1.17i)11-s + (−0.0430 − 0.244i)13-s + (0.0889 − 0.0746i)15-s + (−0.00207 + 0.000754i)17-s + (0.989 − 0.146i)19-s + (0.197 − 0.0718i)21-s + (1.03 − 0.865i)23-s + (0.0347 + 0.196i)25-s + (−0.250 + 0.434i)27-s + (−0.0708 − 0.0257i)29-s + (−0.165 − 0.286i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.773 + 0.633i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.773 + 0.633i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22678 - 0.438313i\)
\(L(\frac12)\) \(\approx\) \(1.22678 - 0.438313i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (-4.31 + 0.640i)T \)
good3 \( 1 + (0.0781 - 0.443i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (1.06 + 1.85i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.25 + 3.90i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.155 + 0.881i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.00854 - 0.00311i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-4.94 + 4.15i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (0.381 + 0.138i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (0.920 + 1.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.75T + 37T^{2} \)
41 \( 1 + (1.24 - 7.04i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-1.04 - 0.880i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-1.13 - 0.413i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-1.90 + 1.59i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-8.35 + 3.04i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (4.44 - 3.72i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (14.6 + 5.34i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (5.71 + 4.79i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (1.93 - 10.9i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (1.27 - 7.25i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.0758 + 0.131i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.98 - 11.2i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (7.85 - 2.85i)T + (74.3 - 62.3i)T^{2} \)
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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

−11.14991870319805255766524532444, −10.40171169294690492430876760246, −9.462461052465032709674733061805, −8.540772769176387001770803732592, −7.44332696906657604894941345705, −6.61644120724053217610765209598, −5.28963227582470402370111960856, −4.18209623229230891325348957680, −3.22062803019955149577220314178, −1.01810927629532665169763914432, 1.66553606194610017337958970968, 3.27308913539597354435502631572, 4.45310863589495463825147721724, 5.72564150636529733739266853293, 7.05249126655435471215769433283, 7.28970619207958993076363165802, 8.899093114686662012386118699942, 9.553299380245475294013692540564, 10.45476251512587561331414031530, 11.78669666291027759581886092750

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