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Properties

Label 2-380-76.27-c1-0-12
Degree $2$
Conductor $380$
Sign $-0.621 - 0.783i$
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  + (0.864 + 1.11i)2-s + (0.0154 − 0.0266i)3-s + (−0.506 + 1.93i)4-s + (−0.5 + 0.866i)5-s + (0.0431 − 0.00580i)6-s − 0.370i·7-s + (−2.60 + 1.10i)8-s + (1.49 + 2.59i)9-s + (−1.40 + 0.188i)10-s + 1.40i·11-s + (0.0438 + 0.0433i)12-s + (−1.25 + 0.727i)13-s + (0.415 − 0.320i)14-s + (0.0154 + 0.0266i)15-s + (−3.48 − 1.96i)16-s + (0.606 − 1.05i)17-s + ⋯
L(s)  = 1  + (0.611 + 0.791i)2-s + (0.00889 − 0.0154i)3-s + (−0.253 + 0.967i)4-s + (−0.223 + 0.387i)5-s + (0.0176 − 0.00237i)6-s − 0.140i·7-s + (−0.920 + 0.390i)8-s + (0.499 + 0.865i)9-s + (−0.443 + 0.0596i)10-s + 0.423i·11-s + (0.0126 + 0.0125i)12-s + (−0.349 + 0.201i)13-s + (0.110 − 0.0856i)14-s + (0.00397 + 0.00688i)15-s + (−0.871 − 0.490i)16-s + (0.147 − 0.254i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.621 - 0.783i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.621 - 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.710101 + 1.47075i\)
\(L(\frac12)\) \(\approx\) \(0.710101 + 1.47075i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.864 - 1.11i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.941 - 4.25i)T \)
good3 \( 1 + (-0.0154 + 0.0266i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 0.370iT - 7T^{2} \)
11 \( 1 - 1.40iT - 11T^{2} \)
13 \( 1 + (1.25 - 0.727i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.606 + 1.05i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.979 + 0.565i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.89 + 1.67i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.52T + 31T^{2} \)
37 \( 1 + 3.60iT - 37T^{2} \)
41 \( 1 + (6.05 + 3.49i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.82 - 4.51i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-9.67 + 5.58i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.26 + 1.30i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.93 - 6.81i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.29 + 2.24i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.17 - 2.03i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.89 + 10.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.94 + 5.09i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.19 - 10.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.54iT - 83T^{2} \)
89 \( 1 + (2.29 - 1.32i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.5 + 6.07i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.98263727455249093432080530821, −10.78066604739197868805679245098, −9.884214853967293330607805037960, −8.627741055645634398342375770912, −7.63093077254725295026161965130, −7.09159693650109349379109516229, −5.93010866904271877371296147015, −4.80410774900090996351447262800, −3.91356377652953167619433236366, −2.43143265934856261422673406153, 0.963332687164689848992336683613, 2.73867598726826895274965508337, 3.92352797256223506507772839321, 4.89250202759716776851965809497, 6.01223255515629680649442508426, 7.05054303000901312162784981524, 8.557460519144919181731078160758, 9.340619632319702159162620292205, 10.24317617652264024695794791482, 11.16789339803849464922633292032

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