Properties

Label 2-380-20.7-c1-0-10
Degree $2$
Conductor $380$
Sign $0.954 - 0.299i$
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  + (−0.282 − 1.38i)2-s + (−0.321 − 0.321i)3-s + (−1.83 + 0.784i)4-s + (1.77 + 1.35i)5-s + (−0.353 + 0.535i)6-s + (−1.91 + 1.91i)7-s + (1.60 + 2.32i)8-s − 2.79i·9-s + (1.37 − 2.84i)10-s + 5.65i·11-s + (0.842 + 0.338i)12-s + (−2.00 + 2.00i)13-s + (3.19 + 2.10i)14-s + (−0.134 − 1.00i)15-s + (2.77 − 2.88i)16-s + (3.89 + 3.89i)17-s + ⋯
L(s)  = 1  + (−0.200 − 0.979i)2-s + (−0.185 − 0.185i)3-s + (−0.919 + 0.392i)4-s + (0.794 + 0.606i)5-s + (−0.144 + 0.218i)6-s + (−0.722 + 0.722i)7-s + (0.568 + 0.822i)8-s − 0.931i·9-s + (0.435 − 0.900i)10-s + 1.70i·11-s + (0.243 + 0.0978i)12-s + (−0.555 + 0.555i)13-s + (0.852 + 0.563i)14-s + (−0.0348 − 0.259i)15-s + (0.692 − 0.721i)16-s + (0.943 + 0.943i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.954 - 0.299i$
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: \((2,\ 380,\ (\ :1/2),\ 0.954 - 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.983099 + 0.150731i\)
\(L(\frac12)\) \(\approx\) \(0.983099 + 0.150731i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.282 + 1.38i)T \)
5 \( 1 + (-1.77 - 1.35i)T \)
19 \( 1 + T \)
good3 \( 1 + (0.321 + 0.321i)T + 3iT^{2} \)
7 \( 1 + (1.91 - 1.91i)T - 7iT^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 + (2.00 - 2.00i)T - 13iT^{2} \)
17 \( 1 + (-3.89 - 3.89i)T + 17iT^{2} \)
23 \( 1 + (-3.04 - 3.04i)T + 23iT^{2} \)
29 \( 1 + 8.56iT - 29T^{2} \)
31 \( 1 + 3.58iT - 31T^{2} \)
37 \( 1 + (-3.87 - 3.87i)T + 37iT^{2} \)
41 \( 1 + 0.0555T + 41T^{2} \)
43 \( 1 + (1.97 + 1.97i)T + 43iT^{2} \)
47 \( 1 + (-4.74 + 4.74i)T - 47iT^{2} \)
53 \( 1 + (7.95 - 7.95i)T - 53iT^{2} \)
59 \( 1 + 8.34T + 59T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 + (6.55 - 6.55i)T - 67iT^{2} \)
71 \( 1 + 7.24iT - 71T^{2} \)
73 \( 1 + (3.34 - 3.34i)T - 73iT^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 + (-4.60 - 4.60i)T + 83iT^{2} \)
89 \( 1 - 4.22iT - 89T^{2} \)
97 \( 1 + (3.74 + 3.74i)T + 97iT^{2} \)
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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

−11.54763350999305607304963977358, −10.25566443913834992242795856675, −9.631128175537482577923532338828, −9.253541730424178718836926069566, −7.67819587169568041375423197955, −6.58855988084356337630976892070, −5.60670152888756258314867413856, −4.17365389933414775934613371015, −2.86394855161479412958587610746, −1.79932955328298139203734671267, 0.76326385178499702357865198015, 3.21839442648448594619837931423, 4.86264048233144003801518957432, 5.48994135939119314443443886324, 6.46965279092532819978452341944, 7.56327504051467557317210380406, 8.507660800308697334397084610682, 9.378905856296349466075007017759, 10.26765174862412001645859713263, 10.89548946310661255409826683468

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