Properties

Label 2-380-380.239-c2-0-58
Degree $2$
Conductor $380$
Sign $0.665 - 0.746i$
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  + (1.73 + 1.00i)2-s + (1.05 − 1.82i)3-s + (1.98 + 3.47i)4-s + (−4.98 − 0.332i)5-s + (3.65 − 2.09i)6-s + 8.55·7-s + (−0.0476 + 7.99i)8-s + (2.28 + 3.95i)9-s + (−8.29 − 5.58i)10-s + 10.2i·11-s + (8.42 + 0.0334i)12-s + (10.2 − 5.89i)13-s + (14.8 + 8.58i)14-s + (−5.86 + 8.75i)15-s + (−8.10 + 13.7i)16-s + (−16.0 − 9.28i)17-s + ⋯
L(s)  = 1  + (0.865 + 0.501i)2-s + (0.351 − 0.608i)3-s + (0.496 + 0.868i)4-s + (−0.997 − 0.0664i)5-s + (0.608 − 0.349i)6-s + 1.22·7-s + (−0.00596 + 0.999i)8-s + (0.253 + 0.438i)9-s + (−0.829 − 0.558i)10-s + 0.928i·11-s + (0.702 + 0.00279i)12-s + (0.785 − 0.453i)13-s + (1.05 + 0.613i)14-s + (−0.390 + 0.583i)15-s + (−0.506 + 0.862i)16-s + (−0.946 − 0.546i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.665 - 0.746i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ 0.665 - 0.746i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.84706 + 1.27564i\)
\(L(\frac12)\) \(\approx\) \(2.84706 + 1.27564i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.73 - 1.00i)T \)
5 \( 1 + (4.98 + 0.332i)T \)
19 \( 1 + (-14.5 + 12.2i)T \)
good3 \( 1 + (-1.05 + 1.82i)T + (-4.5 - 7.79i)T^{2} \)
7 \( 1 - 8.55T + 49T^{2} \)
11 \( 1 - 10.2iT - 121T^{2} \)
13 \( 1 + (-10.2 + 5.89i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (16.0 + 9.28i)T + (144.5 + 250. i)T^{2} \)
23 \( 1 + (-18.2 - 31.6i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (6.36 + 11.0i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + 8.11iT - 961T^{2} \)
37 \( 1 - 43.7iT - 1.36e3T^{2} \)
41 \( 1 + (19.1 - 33.2i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-29.0 + 50.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (21.4 + 37.0i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-1.54 + 0.890i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (68.8 + 39.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (41.5 + 71.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (20.7 + 35.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (79.6 + 45.9i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (85.6 + 49.4i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-15.0 - 8.68i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 144.T + 6.88e3T^{2} \)
89 \( 1 + (9.75 + 16.9i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (14.4 + 8.36i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.48962775537010544166096635173, −10.84495160513587939024794645614, −9.008374463734373881058667440079, −7.964775401662006710104096421050, −7.57893088964576113778890133032, −6.76602391383798356967329196543, −5.07992232343537409192184506241, −4.56336524119195108965252985057, −3.17109850602334378660561432017, −1.71186695638930254697913324111, 1.21640567695521562354843024049, 3.03882938060460235142817049022, 4.05474256162056147144683674081, 4.62148764912932634616469162255, 5.99067102085994736139946495971, 7.15229777040165660659607655182, 8.430397037504644823211665408224, 9.096930874212565208642316715111, 10.69770456651146891021094241337, 10.94045082087941906375117042534

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