Properties

Label 2-380-380.319-c1-0-5
Degree $2$
Conductor $380$
Sign $-0.481 - 0.876i$
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  + (−1.41 + 0.0306i)2-s + (0.151 − 0.0266i)3-s + (1.99 − 0.0867i)4-s + (−0.376 − 2.20i)5-s + (−0.213 + 0.0423i)6-s + (−1.14 + 1.98i)7-s + (−2.82 + 0.183i)8-s + (−2.79 + 1.01i)9-s + (0.599 + 3.10i)10-s + (−2.70 + 1.56i)11-s + (0.299 − 0.0664i)12-s + (−0.571 + 3.23i)13-s + (1.55 − 2.83i)14-s + (−0.115 − 0.323i)15-s + (3.98 − 0.346i)16-s + (0.657 − 1.80i)17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0216i)2-s + (0.0873 − 0.0154i)3-s + (0.999 − 0.0433i)4-s + (−0.168 − 0.985i)5-s + (−0.0869 + 0.0172i)6-s + (−0.432 + 0.748i)7-s + (−0.997 + 0.0650i)8-s + (−0.932 + 0.339i)9-s + (0.189 + 0.981i)10-s + (−0.815 + 0.470i)11-s + (0.0865 − 0.0191i)12-s + (−0.158 + 0.898i)13-s + (0.415 − 0.757i)14-s + (−0.0298 − 0.0835i)15-s + (0.996 − 0.0866i)16-s + (0.159 − 0.438i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.481 - 0.876i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.481 - 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.192496 + 0.325351i\)
\(L(\frac12)\) \(\approx\) \(0.192496 + 0.325351i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0306i)T \)
5 \( 1 + (0.376 + 2.20i)T \)
19 \( 1 + (-1.01 - 4.23i)T \)
good3 \( 1 + (-0.151 + 0.0266i)T + (2.81 - 1.02i)T^{2} \)
7 \( 1 + (1.14 - 1.98i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.70 - 1.56i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.571 - 3.23i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.657 + 1.80i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-6.01 - 5.04i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (1.63 + 4.48i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (2.75 - 4.76i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.87T + 37T^{2} \)
41 \( 1 + (2.91 - 0.514i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-3.18 + 2.66i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (5.85 - 2.13i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-5.37 - 4.51i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (5.48 + 1.99i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (8.49 + 7.13i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (1.09 + 3.01i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (9.98 - 8.38i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-6.42 + 1.13i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.838 + 4.75i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-3.97 + 6.88i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.58 - 0.278i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (-4.18 - 1.52i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.72714500767729456089629616163, −10.64217603993411343462471556955, −9.448012039371633403161744850590, −9.053487810217399554890235302888, −8.104120380552812569295738463505, −7.28602355738078539427456665247, −5.87679403762208500670376828890, −5.09344029331008129882944501282, −3.16495958244331833138966148237, −1.83854243539313508648508677425, 0.32075500842043438691653846905, 2.73072881986342478640199125778, 3.36516761217041179855034833224, 5.51162289606906612710184356208, 6.60110088964930393920912287397, 7.36114406882242110807376558125, 8.271304575093752390418781331543, 9.200853830800596237995939321000, 10.42198513672365259464392733707, 10.70985533593222669142806748908

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