Properties

Label 2-380-1.1-c1-0-2
Degree $2$
Conductor $380$
Sign $-1$
Analytic cond. $3.03431$
Root an. cond. $1.74192$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.41·3-s + 5-s + 0.828·7-s + 8.65·9-s − 2·11-s − 6.24·13-s − 3.41·15-s + 0.828·17-s − 19-s − 2.82·21-s − 6·23-s + 25-s − 19.3·27-s − 6.48·29-s − 6.82·31-s + 6.82·33-s + 0.828·35-s − 1.75·37-s + 21.3·39-s + 3.65·41-s + 4.82·43-s + 8.65·45-s − 4.82·47-s − 6.31·49-s − 2.82·51-s + 9.07·53-s − 2·55-s + ⋯
L(s)  = 1  − 1.97·3-s + 0.447·5-s + 0.313·7-s + 2.88·9-s − 0.603·11-s − 1.73·13-s − 0.881·15-s + 0.200·17-s − 0.229·19-s − 0.617·21-s − 1.25·23-s + 0.200·25-s − 3.71·27-s − 1.20·29-s − 1.22·31-s + 1.18·33-s + 0.140·35-s − 0.288·37-s + 3.41·39-s + 0.571·41-s + 0.736·43-s + 1.29·45-s − 0.704·47-s − 0.901·49-s − 0.396·51-s + 1.24·53-s − 0.269·55-s + ⋯

Functional equation

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

Invariants

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

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 \( 1 - T \)
19 \( 1 + T \)
good3 \( 1 + 3.41T + 3T^{2} \)
7 \( 1 - 0.828T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 6.24T + 13T^{2} \)
17 \( 1 - 0.828T + 17T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 6.48T + 29T^{2} \)
31 \( 1 + 6.82T + 31T^{2} \)
37 \( 1 + 1.75T + 37T^{2} \)
41 \( 1 - 3.65T + 41T^{2} \)
43 \( 1 - 4.82T + 43T^{2} \)
47 \( 1 + 4.82T + 47T^{2} \)
53 \( 1 - 9.07T + 53T^{2} \)
59 \( 1 - 13.6T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 + 3.41T + 67T^{2} \)
71 \( 1 - 5.17T + 71T^{2} \)
73 \( 1 + 2.48T + 73T^{2} \)
79 \( 1 - 1.65T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 + 6.48T + 89T^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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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

−10.94256066319059589801682795120, −10.16623922017043054936950871610, −9.537240522241727969188861774921, −7.69325377850132072651736083877, −6.99267862041749681055782655233, −5.78573202136722975021606241343, −5.26924804531103241711560797133, −4.27638772863881814162798045888, −1.95569357302930458987582739256, 0, 1.95569357302930458987582739256, 4.27638772863881814162798045888, 5.26924804531103241711560797133, 5.78573202136722975021606241343, 6.99267862041749681055782655233, 7.69325377850132072651736083877, 9.537240522241727969188861774921, 10.16623922017043054936950871610, 10.94256066319059589801682795120

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