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Properties

Label 2-380-95.33-c1-0-7
Degree $2$
Conductor $380$
Sign $-0.386 + 0.922i$
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  + (−1.02 − 1.46i)3-s + (0.710 + 2.12i)5-s + (−2.65 − 0.711i)7-s + (−0.0626 + 0.172i)9-s + (2.84 − 4.91i)11-s + (−1.87 − 1.31i)13-s + (2.37 − 3.20i)15-s + (1.57 − 3.38i)17-s + (−3.00 − 3.15i)19-s + (1.67 + 4.60i)21-s + (−3.36 − 0.294i)23-s + (−3.98 + 3.01i)25-s + (−4.85 + 1.30i)27-s + (−1.09 − 0.397i)29-s + (7.63 − 4.40i)31-s + ⋯
L(s)  = 1  + (−0.590 − 0.843i)3-s + (0.317 + 0.948i)5-s + (−1.00 − 0.268i)7-s + (−0.0208 + 0.0573i)9-s + (0.856 − 1.48i)11-s + (−0.520 − 0.364i)13-s + (0.612 − 0.828i)15-s + (0.383 − 0.821i)17-s + (−0.688 − 0.724i)19-s + (0.365 + 1.00i)21-s + (−0.701 − 0.0613i)23-s + (−0.797 + 0.602i)25-s + (−0.934 + 0.250i)27-s + (−0.202 − 0.0738i)29-s + (1.37 − 0.791i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.478948 - 0.720137i\)
\(L(\frac12)\) \(\approx\) \(0.478948 - 0.720137i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.710 - 2.12i)T \)
19 \( 1 + (3.00 + 3.15i)T \)
good3 \( 1 + (1.02 + 1.46i)T + (-1.02 + 2.81i)T^{2} \)
7 \( 1 + (2.65 + 0.711i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-2.84 + 4.91i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.87 + 1.31i)T + (4.44 + 12.2i)T^{2} \)
17 \( 1 + (-1.57 + 3.38i)T + (-10.9 - 13.0i)T^{2} \)
23 \( 1 + (3.36 + 0.294i)T + (22.6 + 3.99i)T^{2} \)
29 \( 1 + (1.09 + 0.397i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-7.63 + 4.40i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.85 - 3.85i)T + 37iT^{2} \)
41 \( 1 + (2.15 + 0.379i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.126 + 1.44i)T + (-42.3 + 7.46i)T^{2} \)
47 \( 1 + (-0.0279 + 0.0130i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (-0.107 + 1.23i)T + (-52.1 - 9.20i)T^{2} \)
59 \( 1 + (-6.00 + 2.18i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-8.16 + 6.85i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-5.85 - 12.5i)T + (-43.0 + 51.3i)T^{2} \)
71 \( 1 + (7.74 - 9.23i)T + (-12.3 - 69.9i)T^{2} \)
73 \( 1 + (-10.1 + 7.08i)T + (24.9 - 68.5i)T^{2} \)
79 \( 1 + (2.15 - 12.2i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-3.01 + 11.2i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-1.92 - 10.9i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (10.5 + 4.90i)T + (62.3 + 74.3i)T^{2} \)
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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

−11.34427277185883237848605805990, −10.14220357099175432318970118173, −9.456705075022835690238884589680, −8.096167112423166493597137078427, −6.85757866994547765601872942126, −6.49910991122169305565003475672, −5.66315789786761989738965322188, −3.75338700392175617209740047143, −2.66261243255107460633512707872, −0.61755309218295755181859545709, 1.92463692056796038262682787738, 3.97429651301759765680925191660, 4.65769625539116730527932815027, 5.77381424641051426227586584204, 6.65422689153214200186799906953, 8.051902717712391398834257398817, 9.270207441466392286173224344921, 9.838932320895754165852520418189, 10.38124881907858099153773438895, 11.87908218938353398110164334800

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