Properties

Label 2-380-380.219-c1-0-32
Degree $2$
Conductor $380$
Sign $-0.739 + 0.672i$
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  + (−0.934 − 1.06i)2-s + (−0.912 − 1.08i)3-s + (−0.251 + 1.98i)4-s + (1.86 − 1.22i)5-s + (−0.300 + 1.98i)6-s + (−0.869 + 1.50i)7-s + (2.34 − 1.58i)8-s + (0.170 − 0.969i)9-s + (−3.05 − 0.832i)10-s + (1.98 − 1.14i)11-s + (2.38 − 1.53i)12-s + (−1.18 − 0.990i)13-s + (2.41 − 0.485i)14-s + (−3.04 − 0.909i)15-s + (−3.87 − 0.998i)16-s + (5.38 − 0.949i)17-s + ⋯
L(s)  = 1  + (−0.661 − 0.750i)2-s + (−0.526 − 0.627i)3-s + (−0.125 + 0.992i)4-s + (0.835 − 0.549i)5-s + (−0.122 + 0.810i)6-s + (−0.328 + 0.569i)7-s + (0.827 − 0.561i)8-s + (0.0569 − 0.323i)9-s + (−0.964 − 0.263i)10-s + (0.599 − 0.346i)11-s + (0.689 − 0.443i)12-s + (−0.327 − 0.274i)13-s + (0.644 − 0.129i)14-s + (−0.785 − 0.234i)15-s + (−0.968 − 0.249i)16-s + (1.30 − 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 + 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.739 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.739 + 0.672i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.739 + 0.672i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.305052 - 0.788659i\)
\(L(\frac12)\) \(\approx\) \(0.305052 - 0.788659i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.934 + 1.06i)T \)
5 \( 1 + (-1.86 + 1.22i)T \)
19 \( 1 + (0.368 + 4.34i)T \)
good3 \( 1 + (0.912 + 1.08i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (0.869 - 1.50i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.98 + 1.14i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.18 + 0.990i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-5.38 + 0.949i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (2.44 - 0.891i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (8.70 + 1.53i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (-2.00 + 3.47i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.47T + 37T^{2} \)
41 \( 1 + (-3.48 - 4.14i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (2.41 + 0.880i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.38 + 7.84i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (1.36 - 0.495i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.46 + 8.28i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-6.17 + 2.24i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-10.5 - 1.85i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-5.69 - 2.07i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-4.66 - 5.56i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-0.266 + 0.223i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-3.33 + 5.77i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.63 - 7.90i)T + (-15.4 - 87.6i)T^{2} \)
97 \( 1 + (0.978 + 5.54i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.18084262497155383888173150075, −9.798835595493309893050991337853, −9.460156783107472408534827078792, −8.457716721402439465927862590719, −7.29662165551656066502051561340, −6.23905515436315623189441514445, −5.27525581331269107742156061628, −3.59073305268069915943483976659, −2.14039345636456507161932113688, −0.78630041630665336820885116000, 1.75838076196794036502623684563, 3.87697804059311821646273566003, 5.24176933141250094743139252283, 5.97094956191671267948340605041, 6.98354815889879525066661663236, 7.81570591338111919056177818995, 9.212180333369425255931475805070, 10.04706096279471758028103211494, 10.31788754694799357603332401135, 11.27896547324414958515861654009

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