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Properties

Label 2-380-19.17-c1-0-4
Degree $2$
Conductor $380$
Sign $0.281 + 0.959i$
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  + (−0.679 − 0.569i)3-s + (−0.939 + 0.342i)5-s + (0.460 + 0.797i)7-s + (−0.384 − 2.18i)9-s + (1.03 − 1.79i)11-s + (3.22 − 2.70i)13-s + (0.833 + 0.303i)15-s + (0.340 − 1.93i)17-s + (0.207 − 4.35i)19-s + (0.141 − 0.803i)21-s + (−0.570 − 0.207i)23-s + (0.766 − 0.642i)25-s + (−2.31 + 4.00i)27-s + (−1.31 − 7.44i)29-s + (1.07 + 1.86i)31-s + ⋯
L(s)  = 1  + (−0.392 − 0.328i)3-s + (−0.420 + 0.152i)5-s + (0.174 + 0.301i)7-s + (−0.128 − 0.726i)9-s + (0.313 − 0.542i)11-s + (0.894 − 0.750i)13-s + (0.215 + 0.0782i)15-s + (0.0825 − 0.468i)17-s + (0.0476 − 0.998i)19-s + (0.0309 − 0.175i)21-s + (−0.118 − 0.0432i)23-s + (0.153 − 0.128i)25-s + (−0.444 + 0.770i)27-s + (−0.243 − 1.38i)29-s + (0.192 + 0.334i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.281 + 0.959i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.281 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.850917 - 0.636967i\)
\(L(\frac12)\) \(\approx\) \(0.850917 - 0.636967i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (-0.207 + 4.35i)T \)
good3 \( 1 + (0.679 + 0.569i)T + (0.520 + 2.95i)T^{2} \)
7 \( 1 + (-0.460 - 0.797i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.03 + 1.79i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.22 + 2.70i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.340 + 1.93i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (0.570 + 0.207i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (1.31 + 7.44i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-1.07 - 1.86i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.88T + 37T^{2} \)
41 \( 1 + (4.50 + 3.77i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (0.638 - 0.232i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-1.88 - 10.6i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-4.25 - 1.54i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (1.02 - 5.82i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-6.55 - 2.38i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.32 - 7.52i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (10.8 - 3.93i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-2.42 - 2.03i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (2.36 + 1.98i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (5.06 + 8.76i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.60 + 2.18i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (3.29 - 18.7i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.48572154704719586131726556355, −10.41786964428208208488527162853, −9.193642660563882423194589384577, −8.431775819383861194043382127150, −7.34017566020637240838031451061, −6.32428079037144137425954689775, −5.53497363675876333637528312296, −4.06328552613076766285600188514, −2.88692564701645497947256037334, −0.808720931616163485852469047015, 1.71061334172644079882253470308, 3.68152744621448474159355822027, 4.56123789008799641978528307797, 5.66597810562335628170978957346, 6.80557617437231525153818217898, 7.88260796180698860642522148538, 8.704560954862953622417552026315, 9.877613646519752552945285654783, 10.71840633295734712143307080810, 11.44271802123270396038043012309

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