Properties

Label 2-380-380.239-c2-0-41
Degree $2$
Conductor $380$
Sign $0.930 + 0.366i$
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  + (−0.691 − 1.87i)2-s + (1.97 − 3.42i)3-s + (−3.04 + 2.59i)4-s + (−0.693 + 4.95i)5-s + (−7.78 − 1.34i)6-s + 2.56·7-s + (6.97 + 3.91i)8-s + (−3.30 − 5.73i)9-s + (9.77 − 2.12i)10-s + 17.7i·11-s + (2.87 + 15.5i)12-s + (−4.58 + 2.64i)13-s + (−1.77 − 4.82i)14-s + (15.5 + 12.1i)15-s + (2.52 − 15.8i)16-s + (1.90 + 1.09i)17-s + ⋯
L(s)  = 1  + (−0.345 − 0.938i)2-s + (0.658 − 1.14i)3-s + (−0.760 + 0.649i)4-s + (−0.138 + 0.990i)5-s + (−1.29 − 0.223i)6-s + 0.367·7-s + (0.872 + 0.489i)8-s + (−0.367 − 0.636i)9-s + (0.977 − 0.212i)10-s + 1.61i·11-s + (0.239 + 1.29i)12-s + (−0.352 + 0.203i)13-s + (−0.126 − 0.344i)14-s + (1.03 + 0.810i)15-s + (0.157 − 0.987i)16-s + (0.111 + 0.0645i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.366i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.930 + 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.930 + 0.366i$
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: \((2,\ 380,\ (\ :1),\ 0.930 + 0.366i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.52002 - 0.288940i\)
\(L(\frac12)\) \(\approx\) \(1.52002 - 0.288940i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.691 + 1.87i)T \)
5 \( 1 + (0.693 - 4.95i)T \)
19 \( 1 + (-15.3 - 11.2i)T \)
good3 \( 1 + (-1.97 + 3.42i)T + (-4.5 - 7.79i)T^{2} \)
7 \( 1 - 2.56T + 49T^{2} \)
11 \( 1 - 17.7iT - 121T^{2} \)
13 \( 1 + (4.58 - 2.64i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-1.90 - 1.09i)T + (144.5 + 250. i)T^{2} \)
23 \( 1 + (-8.21 - 14.2i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-24.5 - 42.4i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + 13.0iT - 961T^{2} \)
37 \( 1 + 53.5iT - 1.36e3T^{2} \)
41 \( 1 + (-12.6 + 21.9i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (36.7 - 63.7i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (27.9 + 48.4i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-38.6 + 22.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (58.4 + 33.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (2.07 + 3.59i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-51.0 - 88.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (35.9 + 20.7i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-41.1 - 23.7i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-63.0 - 36.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 124.T + 6.88e3T^{2} \)
89 \( 1 + (-50.2 - 86.9i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (53.4 + 30.8i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.16069364504058666861411742200, −10.14882138216880698081027451748, −9.413783323482327701498970673607, −8.146744388774735115834235848045, −7.45659058905465785244992158004, −6.87654006418756171039658108334, −4.96291365104400537136996076533, −3.53163316269012690058475931183, −2.39737371479319071294626245627, −1.55843502068957584803868387402, 0.78293340970658221839352945643, 3.26144281652791710027167799728, 4.53564288460965148060252154603, 5.14278738536007614657073294138, 6.32582768179549409198136375687, 7.917484456961575795638638365391, 8.454563913605533235318844192814, 9.166721342498351714574602509444, 9.903130102786076759308810523006, 10.87871905329496373235349046849

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