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Properties

Label 2-380-95.13-c1-0-6
Degree $2$
Conductor $380$
Sign $0.998 + 0.0621i$
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  + (3.00 − 1.39i)3-s + (−0.904 + 2.04i)5-s + (1.03 + 3.87i)7-s + (5.12 − 6.10i)9-s + (−1.27 − 2.20i)11-s + (0.897 − 1.92i)13-s + (0.146 + 7.40i)15-s + (−0.521 + 5.96i)17-s + (−1.63 − 4.03i)19-s + (8.53 + 10.1i)21-s + (−0.709 + 0.497i)23-s + (−3.36 − 3.69i)25-s + (4.25 − 15.8i)27-s + (0.706 + 0.592i)29-s + (−3.26 − 1.88i)31-s + ⋯
L(s)  = 1  + (1.73 − 0.808i)3-s + (−0.404 + 0.914i)5-s + (0.392 + 1.46i)7-s + (1.70 − 2.03i)9-s + (−0.384 − 0.665i)11-s + (0.248 − 0.533i)13-s + (0.0379 + 1.91i)15-s + (−0.126 + 1.44i)17-s + (−0.375 − 0.926i)19-s + (1.86 + 2.21i)21-s + (−0.148 + 0.103i)23-s + (−0.672 − 0.739i)25-s + (0.819 − 3.05i)27-s + (0.131 + 0.110i)29-s + (−0.586 − 0.338i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.998 + 0.0621i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.998 + 0.0621i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22959 - 0.0693237i\)
\(L(\frac12)\) \(\approx\) \(2.22959 - 0.0693237i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.904 - 2.04i)T \)
19 \( 1 + (1.63 + 4.03i)T \)
good3 \( 1 + (-3.00 + 1.39i)T + (1.92 - 2.29i)T^{2} \)
7 \( 1 + (-1.03 - 3.87i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.27 + 2.20i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.897 + 1.92i)T + (-8.35 - 9.95i)T^{2} \)
17 \( 1 + (0.521 - 5.96i)T + (-16.7 - 2.95i)T^{2} \)
23 \( 1 + (0.709 - 0.497i)T + (7.86 - 21.6i)T^{2} \)
29 \( 1 + (-0.706 - 0.592i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (3.26 + 1.88i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.20 + 4.20i)T + 37iT^{2} \)
41 \( 1 + (0.681 - 1.87i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (-4.59 + 6.56i)T + (-14.7 - 40.4i)T^{2} \)
47 \( 1 + (5.41 - 0.473i)T + (46.2 - 8.16i)T^{2} \)
53 \( 1 + (0.560 + 0.800i)T + (-18.1 + 49.8i)T^{2} \)
59 \( 1 + (0.574 - 0.482i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.540 + 3.06i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-1.15 - 13.2i)T + (-65.9 + 11.6i)T^{2} \)
71 \( 1 + (7.53 - 1.32i)T + (66.7 - 24.2i)T^{2} \)
73 \( 1 + (-0.911 - 1.95i)T + (-46.9 + 55.9i)T^{2} \)
79 \( 1 + (-10.7 - 3.92i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (6.50 - 1.74i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (8.47 - 3.08i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-0.658 - 0.0576i)T + (95.5 + 16.8i)T^{2} \)
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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

−11.42594345076557992441367485090, −10.38397738125114866180601529736, −9.080678571666436312124019461275, −8.458308715779274779917145259167, −7.897811144013453318905655428131, −6.82143954358203461987627872188, −5.80185956701487043184913671058, −3.82225158977742926693692685107, −2.85529934015158696937453011008, −2.04932218991607266715336089292, 1.73081186960900252390421456760, 3.37363205871510547471607936343, 4.34758528253529128245538290577, 4.82153196455685716366677180323, 7.20354654163680329006557865320, 7.79072467774946304667355093067, 8.598885193513867635401348469308, 9.503412647257342978133380695240, 10.14729937800783357962132774850, 11.11633656064731711151826635345

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