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Properties

Label 2-380-380.379-c1-0-41
Degree $2$
Conductor $380$
Sign $-0.990 + 0.139i$
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  + (−0.584 − 1.28i)2-s + 2.81i·3-s + (−1.31 + 1.50i)4-s + (0.390 − 2.20i)5-s + (3.62 − 1.64i)6-s − 4.13·7-s + (2.70 + 0.813i)8-s − 4.91·9-s + (−3.06 + 0.785i)10-s + 0.681i·11-s + (−4.23 − 3.70i)12-s − 3.19·13-s + (2.41 + 5.31i)14-s + (6.19 + 1.09i)15-s + (−0.535 − 3.96i)16-s − 2.93i·17-s + ⋯
L(s)  = 1  + (−0.413 − 0.910i)2-s + 1.62i·3-s + (−0.658 + 0.752i)4-s + (0.174 − 0.984i)5-s + (1.47 − 0.671i)6-s − 1.56·7-s + (0.957 + 0.287i)8-s − 1.63·9-s + (−0.968 + 0.248i)10-s + 0.205i·11-s + (−1.22 − 1.06i)12-s − 0.886·13-s + (0.645 + 1.42i)14-s + (1.59 + 0.283i)15-s + (−0.133 − 0.990i)16-s − 0.711i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.990 + 0.139i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.990 + 0.139i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00431132 - 0.0616991i\)
\(L(\frac12)\) \(\approx\) \(0.00431132 - 0.0616991i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.584 + 1.28i)T \)
5 \( 1 + (-0.390 + 2.20i)T \)
19 \( 1 + (-2.23 + 3.74i)T \)
good3 \( 1 - 2.81iT - 3T^{2} \)
7 \( 1 + 4.13T + 7T^{2} \)
11 \( 1 - 0.681iT - 11T^{2} \)
13 \( 1 + 3.19T + 13T^{2} \)
17 \( 1 + 2.93iT - 17T^{2} \)
23 \( 1 + 6.41T + 23T^{2} \)
29 \( 1 + 0.928iT - 29T^{2} \)
31 \( 1 + 6.68T + 31T^{2} \)
37 \( 1 + 6.63T + 37T^{2} \)
41 \( 1 - 4.89iT - 41T^{2} \)
43 \( 1 - 7.02T + 43T^{2} \)
47 \( 1 - 6.74T + 47T^{2} \)
53 \( 1 - 7.60T + 53T^{2} \)
59 \( 1 + 13.1T + 59T^{2} \)
61 \( 1 + 5.76T + 61T^{2} \)
67 \( 1 - 5.12iT - 67T^{2} \)
71 \( 1 + 7.43T + 71T^{2} \)
73 \( 1 - 10.1iT - 73T^{2} \)
79 \( 1 - 4.94T + 79T^{2} \)
83 \( 1 + 9.74T + 83T^{2} \)
89 \( 1 - 7.65iT - 89T^{2} \)
97 \( 1 - 15.0T + 97T^{2} \)
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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

−10.57493367142886287640293656009, −9.834861367329307407183327306548, −9.398337342062875741631978234697, −8.871222737408494899458450414037, −7.39491116994841391815997532753, −5.61574768158732059182249657940, −4.64585650035222978904990357578, −3.77897394163288189032860795245, −2.67470223302909644210606579922, −0.04428871817572210493208776890, 2.02490544825383519694738903059, 3.53089551586869608790872926608, 5.93792071213841544265228058996, 6.11566241638946222654912994323, 7.31610803904421700450623151979, 7.46574864109800205202288385281, 8.842482068433796673166059393923, 9.884566460498930563138071954937, 10.61161660527192095010788347590, 12.15806986685166154881536855695

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