Properties

Label 2-380-380.7-c1-0-31
Degree $2$
Conductor $380$
Sign $0.925 - 0.378i$
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  + (−1.37 + 0.341i)2-s + (−0.280 + 1.04i)3-s + (1.76 − 0.936i)4-s + (1.80 + 1.31i)5-s + (0.0279 − 1.52i)6-s + (3.16 − 3.16i)7-s + (−2.10 + 1.88i)8-s + (1.58 + 0.914i)9-s + (−2.92 − 1.19i)10-s − 0.437i·11-s + (0.483 + 2.10i)12-s + (−1.27 − 4.76i)13-s + (−3.26 + 5.41i)14-s + (−1.88 + 1.51i)15-s + (2.24 − 3.30i)16-s + (1.14 − 4.27i)17-s + ⋯
L(s)  = 1  + (−0.970 + 0.241i)2-s + (−0.161 + 0.603i)3-s + (0.883 − 0.468i)4-s + (0.807 + 0.589i)5-s + (0.0114 − 0.624i)6-s + (1.19 − 1.19i)7-s + (−0.744 + 0.667i)8-s + (0.528 + 0.304i)9-s + (−0.926 − 0.377i)10-s − 0.131i·11-s + (0.139 + 0.608i)12-s + (−0.354 − 1.32i)13-s + (−0.871 + 1.44i)14-s + (−0.486 + 0.392i)15-s + (0.561 − 0.827i)16-s + (0.278 − 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.925 - 0.378i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.925 - 0.378i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10531 + 0.217037i\)
\(L(\frac12)\) \(\approx\) \(1.10531 + 0.217037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.341i)T \)
5 \( 1 + (-1.80 - 1.31i)T \)
19 \( 1 + (-1.42 + 4.11i)T \)
good3 \( 1 + (0.280 - 1.04i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-3.16 + 3.16i)T - 7iT^{2} \)
11 \( 1 + 0.437iT - 11T^{2} \)
13 \( 1 + (1.27 + 4.76i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-1.14 + 4.27i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (5.89 - 1.58i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-7.30 - 4.21i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.09iT - 31T^{2} \)
37 \( 1 + (-0.393 - 0.393i)T + 37iT^{2} \)
41 \( 1 + (3.07 + 5.31i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.34 - 5.00i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (0.159 + 0.593i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.558 - 2.08i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.33 - 7.50i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.13 - 12.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.44 - 5.37i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (7.25 - 4.18i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.26 + 2.21i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.85 + 8.40i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.249 + 0.249i)T + 83iT^{2} \)
89 \( 1 + (9.89 + 5.71i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.38 + 5.16i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.91251435754649923251402403840, −10.29097738428986622013805921390, −9.989071684594548734257803882007, −8.685453152972810193570761704378, −7.50673989730210302105715543976, −7.11252643785528649564581129476, −5.58503368706172771499639069363, −4.74597976333233389311620760866, −2.91471935650641108935240985277, −1.29119348318967895215600372186, 1.61099316459458801510332945836, 2.09468910397482644846071391258, 4.34903424746546717959757101310, 5.83788883332043200581445078907, 6.52625015453674842944933359086, 7.951167934584353347402366835589, 8.434374821085801361134080014727, 9.520171759121605812086600676733, 10.09763299025305810131192056526, 11.49008267501021303900188717306

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