Properties

Label 2-380-380.239-c2-0-66
Degree $2$
Conductor $380$
Sign $-0.591 + 0.806i$
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  + (−1.92 − 0.532i)2-s + (−1.00 + 1.74i)3-s + (3.43 + 2.05i)4-s + (−4.76 − 1.51i)5-s + (2.87 − 2.83i)6-s + 3.07·7-s + (−5.52 − 5.78i)8-s + (2.46 + 4.26i)9-s + (8.37 + 5.45i)10-s − 3.77i·11-s + (−7.05 + 3.93i)12-s + (−9.11 + 5.26i)13-s + (−5.92 − 1.63i)14-s + (7.46 − 6.80i)15-s + (7.58 + 14.0i)16-s + (9.93 + 5.73i)17-s + ⋯
L(s)  = 1  + (−0.963 − 0.266i)2-s + (−0.336 + 0.582i)3-s + (0.858 + 0.512i)4-s + (−0.952 − 0.303i)5-s + (0.479 − 0.472i)6-s + 0.439·7-s + (−0.691 − 0.722i)8-s + (0.273 + 0.473i)9-s + (0.837 + 0.545i)10-s − 0.342i·11-s + (−0.587 + 0.327i)12-s + (−0.701 + 0.404i)13-s + (−0.423 − 0.116i)14-s + (0.497 − 0.453i)15-s + (0.473 + 0.880i)16-s + (0.584 + 0.337i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 + 0.806i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.591 + 0.806i$
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: \((2,\ 380,\ (\ :1),\ -0.591 + 0.806i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.100371 - 0.198159i\)
\(L(\frac12)\) \(\approx\) \(0.100371 - 0.198159i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.92 + 0.532i)T \)
5 \( 1 + (4.76 + 1.51i)T \)
19 \( 1 + (16.4 - 9.51i)T \)
good3 \( 1 + (1.00 - 1.74i)T + (-4.5 - 7.79i)T^{2} \)
7 \( 1 - 3.07T + 49T^{2} \)
11 \( 1 + 3.77iT - 121T^{2} \)
13 \( 1 + (9.11 - 5.26i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-9.93 - 5.73i)T + (144.5 + 250. i)T^{2} \)
23 \( 1 + (6.96 + 12.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (9.09 + 15.7i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + 50.1iT - 961T^{2} \)
37 \( 1 - 17.0iT - 1.36e3T^{2} \)
41 \( 1 + (-35.7 + 61.9i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (2.56 - 4.44i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (27.8 + 48.1i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-21.0 + 12.1i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (79.5 + 45.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (2.51 + 4.36i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (16.9 + 29.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (112. + 65.1i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (69.2 + 39.9i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (6.25 + 3.61i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 61.2T + 6.88e3T^{2} \)
89 \( 1 + (-14.2 - 24.6i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-19.9 - 11.5i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.74793552971513951278586809979, −10.03630727743306458637833568114, −9.002959382141342053512741448686, −8.002164020507361432946724016727, −7.51671496444291314092537778743, −6.08713818067702352037936475084, −4.66469016065643106879330365003, −3.73399418475127228731811951590, −2.00984765040052041979944213504, −0.14920166395662620282710068797, 1.32752375257938539818134807721, 2.99263387106191383833690091304, 4.68833951402390264519583662908, 6.05618714352539237145127503847, 7.14131108228556338909522901452, 7.52660199676943870216943723331, 8.532573170015402269607986421615, 9.595863019217801008506402407980, 10.57662820969865292844326134876, 11.40871195537492059697726178691

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