Properties

Label 2-380-76.7-c2-0-4
Degree $2$
Conductor $380$
Sign $-0.820 - 0.570i$
Analytic cond. $10.3542$
Root an. cond. $3.21780$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.98 + 0.270i)2-s + (−0.144 + 0.0837i)3-s + (3.85 − 1.07i)4-s + (1.11 + 1.93i)5-s + (0.264 − 0.205i)6-s − 9.41i·7-s + (−7.34 + 3.16i)8-s + (−4.48 + 7.76i)9-s + (−2.73 − 3.53i)10-s − 10.2i·11-s + (−0.468 + 0.477i)12-s + (−11.3 + 19.7i)13-s + (2.54 + 18.6i)14-s + (−0.324 − 0.187i)15-s + (13.7 − 8.26i)16-s + (−4.16 − 7.22i)17-s + ⋯
L(s)  = 1  + (−0.990 + 0.135i)2-s + (−0.0483 + 0.0279i)3-s + (0.963 − 0.267i)4-s + (0.223 + 0.387i)5-s + (0.0441 − 0.0341i)6-s − 1.34i·7-s + (−0.918 + 0.395i)8-s + (−0.498 + 0.863i)9-s + (−0.273 − 0.353i)10-s − 0.932i·11-s + (−0.0390 + 0.0398i)12-s + (−0.875 + 1.51i)13-s + (0.181 + 1.33i)14-s + (−0.0216 − 0.0124i)15-s + (0.856 − 0.516i)16-s + (−0.245 − 0.424i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 - 0.570i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.820 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.820 - 0.570i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ -0.820 - 0.570i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.107389 + 0.342481i\)
\(L(\frac12)\) \(\approx\) \(0.107389 + 0.342481i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.98 - 0.270i)T \)
5 \( 1 + (-1.11 - 1.93i)T \)
19 \( 1 + (2.69 - 18.8i)T \)
good3 \( 1 + (0.144 - 0.0837i)T + (4.5 - 7.79i)T^{2} \)
7 \( 1 + 9.41iT - 49T^{2} \)
11 \( 1 + 10.2iT - 121T^{2} \)
13 \( 1 + (11.3 - 19.7i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (4.16 + 7.22i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-2.10 - 1.21i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-8.50 + 14.7i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 - 26.6iT - 961T^{2} \)
37 \( 1 + 35.6T + 1.36e3T^{2} \)
41 \( 1 + (-30.0 - 52.0i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (64.6 - 37.3i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (26.0 + 15.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (2.71 - 4.70i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (53.5 - 30.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (55.3 - 95.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (41.9 + 24.2i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-22.2 + 12.8i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (30.1 + 52.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (70.3 - 40.5i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 13.8iT - 6.88e3T^{2} \)
89 \( 1 + (-25.5 + 44.1i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-73.1 - 126. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.22961687586218487150000389652, −10.49374558132674708811473083709, −9.813049209498382835293230847408, −8.722762262549491060128382020774, −7.76016023721261634445894026360, −6.98996807466353180470726997456, −6.09772172313963627064616618537, −4.65624398883837750746971774775, −3.06238076517849818615091365681, −1.62614315799412598006373396413, 0.20823085313304104751155903374, 2.08891877624499800027792053361, 3.10458283738330581081995805009, 5.11201737996536608766168241750, 6.03579714115884937485216109709, 7.11744440652168546136149839885, 8.265619722701158759293707786257, 8.995039680955604875549309124187, 9.659438267610475718940087214491, 10.59585896143174565034407344443

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