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Properties

Label 2-380-19.11-c1-0-6
Degree $2$
Conductor $380$
Sign $-0.813 + 0.582i$
Analytic cond. $3.03431$
Root an. cond. $1.74192$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)3-s + (0.5 − 0.866i)5-s − 4·7-s + (−0.499 − 0.866i)9-s − 3·11-s + (−3 − 5.19i)13-s + (0.999 + 1.73i)15-s + (−1 + 1.73i)17-s + (3.5 + 2.59i)19-s + (4 − 6.92i)21-s + (−2 − 3.46i)23-s + (−0.499 − 0.866i)25-s − 4.00·27-s + (−0.5 − 0.866i)29-s − 5·31-s + ⋯
L(s)  = 1  + (−0.577 + 0.999i)3-s + (0.223 − 0.387i)5-s − 1.51·7-s + (−0.166 − 0.288i)9-s − 0.904·11-s + (−0.832 − 1.44i)13-s + (0.258 + 0.447i)15-s + (−0.242 + 0.420i)17-s + (0.802 + 0.596i)19-s + (0.872 − 1.51i)21-s + (−0.417 − 0.722i)23-s + (−0.0999 − 0.173i)25-s − 0.769·27-s + (−0.0928 − 0.160i)29-s − 0.898·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.813 + 0.582i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.813 + 0.582i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-3.5 - 2.59i)T \)
good3 \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 4T + 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7 - 12.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.5 - 12.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6 - 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16T + 83T^{2} \)
89 \( 1 + (8.5 + 14.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6 - 10.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.55005871258609231096417376421, −10.15811909647451863089837031259, −9.567657578894916913397918485686, −8.299094464394035898817830798052, −7.15436654659483222524648393740, −5.74931181956667392369599391798, −5.30748467487694087403098484756, −3.95831158100174324999440662272, −2.78444715066836715824229707174, 0, 2.11662701960827812964030665924, 3.42170408054527542682350484218, 5.14128687624197029502197343284, 6.26236162132795420665652425092, 6.94346164915925597773757949648, 7.52991412459859746622069974894, 9.290062157890664337851738653811, 9.724928409640177781548222006964, 10.95631462724584170183734461757

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