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Properties

Label 2-380-19.7-c1-0-3
Degree $2$
Conductor $380$
Sign $0.257 + 0.966i$
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  + (−1.28 − 2.22i)3-s + (0.5 + 0.866i)5-s + 3.56·7-s + (−1.79 + 3.11i)9-s + 4.56·11-s + (0.5 − 0.866i)13-s + (1.28 − 2.22i)15-s + (−2.86 − 4.96i)17-s + (−4.35 − 0.221i)19-s + (−4.58 − 7.93i)21-s + (2.79 − 4.84i)23-s + (−0.499 + 0.866i)25-s + 1.53·27-s + (−3.38 + 5.86i)29-s + 4.59·31-s + ⋯
L(s)  = 1  + (−0.741 − 1.28i)3-s + (0.223 + 0.387i)5-s + 1.34·7-s + (−0.599 + 1.03i)9-s + 1.37·11-s + (0.138 − 0.240i)13-s + (0.331 − 0.574i)15-s + (−0.695 − 1.20i)17-s + (−0.998 − 0.0507i)19-s + (−1.00 − 1.73i)21-s + (0.583 − 1.01i)23-s + (−0.0999 + 0.173i)25-s + 0.296·27-s + (−0.628 + 1.08i)29-s + 0.826·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.257 + 0.966i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.257 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02386 - 0.787156i\)
\(L(\frac12)\) \(\approx\) \(1.02386 - 0.787156i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.35 + 0.221i)T \)
good3 \( 1 + (1.28 + 2.22i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 3.56T + 7T^{2} \)
11 \( 1 - 4.56T + 11T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.86 + 4.96i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.79 + 4.84i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.38 - 5.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.59T + 31T^{2} \)
37 \( 1 - 3.56T + 37T^{2} \)
41 \( 1 + (5.35 + 9.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.65 - 9.79i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.38 + 5.86i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.70 + 2.94i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.38 - 5.86i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.06 - 7.04i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.16 + 5.48i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.515 - 0.892i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.36 - 11.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.43 + 4.21i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.40T + 83T^{2} \)
89 \( 1 + (6.13 - 10.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.31 + 2.27i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.32612383946419793054440356927, −10.69302062063438493910000038724, −9.136451147427342885022343472573, −8.254731377448763131822700957208, −7.07056483798666799673964955081, −6.65095894719626265797734191094, −5.49946840747898845250867452852, −4.35574261326355407857798510147, −2.32405093881321929532957298535, −1.12327458086508999284962222690, 1.66428680258569218101136840898, 4.04839693833413444013890709896, 4.45555704964659575626871601899, 5.58496707864959900837781887009, 6.48193087252934217349618102331, 8.084681149750119417481130241823, 8.966576015810225007775906846066, 9.741129483591832553440506499686, 10.85012116658856642149587663509, 11.29825470371034595919765580811

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