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Properties

Label 2-380-95.37-c1-0-6
Degree $2$
Conductor $380$
Sign $0.946 - 0.321i$
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  + (1.49 + 1.49i)3-s + (0.311 − 2.21i)5-s + (0.311 + 0.311i)7-s + 1.47i·9-s + 2.90·11-s + (2.84 + 2.84i)13-s + (3.77 − 2.84i)15-s + (−2.52 − 2.52i)17-s + (4.34 + 0.377i)19-s + 0.930i·21-s + (−4.11 + 4.11i)23-s + (−4.80 − 1.37i)25-s + (2.28 − 2.28i)27-s − 2.99·29-s + 0.930i·31-s + ⋯
L(s)  = 1  + (0.863 + 0.863i)3-s + (0.139 − 0.990i)5-s + (0.117 + 0.117i)7-s + 0.491i·9-s + 0.875·11-s + (0.789 + 0.789i)13-s + (0.975 − 0.735i)15-s + (−0.612 − 0.612i)17-s + (0.996 + 0.0866i)19-s + 0.203i·21-s + (−0.858 + 0.858i)23-s + (−0.961 − 0.275i)25-s + (0.439 − 0.439i)27-s − 0.555·29-s + 0.167i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.946 - 0.321i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.946 - 0.321i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87482 + 0.309498i\)
\(L(\frac12)\) \(\approx\) \(1.87482 + 0.309498i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.311 + 2.21i)T \)
19 \( 1 + (-4.34 - 0.377i)T \)
good3 \( 1 + (-1.49 - 1.49i)T + 3iT^{2} \)
7 \( 1 + (-0.311 - 0.311i)T + 7iT^{2} \)
11 \( 1 - 2.90T + 11T^{2} \)
13 \( 1 + (-2.84 - 2.84i)T + 13iT^{2} \)
17 \( 1 + (2.52 + 2.52i)T + 17iT^{2} \)
23 \( 1 + (4.11 - 4.11i)T - 23iT^{2} \)
29 \( 1 + 2.99T + 29T^{2} \)
31 \( 1 - 0.930iT - 31T^{2} \)
37 \( 1 + (8.11 - 8.11i)T - 37iT^{2} \)
41 \( 1 + 2.06iT - 41T^{2} \)
43 \( 1 + (-2.11 + 2.11i)T - 43iT^{2} \)
47 \( 1 + (2.73 + 2.73i)T + 47iT^{2} \)
53 \( 1 + (-0.565 - 0.565i)T + 53iT^{2} \)
59 \( 1 + 9.32T + 59T^{2} \)
61 \( 1 - 3.52T + 61T^{2} \)
67 \( 1 + (-0.144 + 0.144i)T - 67iT^{2} \)
71 \( 1 + 9.61iT - 71T^{2} \)
73 \( 1 + (4.09 - 4.09i)T - 73iT^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 + (-9.21 + 9.21i)T - 83iT^{2} \)
89 \( 1 - 7.55T + 89T^{2} \)
97 \( 1 + (12.0 - 12.0i)T - 97iT^{2} \)
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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

−11.59421806959456022965839840560, −10.18446907198078956651089308425, −9.230954044759049940608905190999, −9.015935089823080649086794631735, −8.018739544415010127505799670995, −6.63831785832611446270634830146, −5.33862731309666759694224182099, −4.26901915354712035221402980350, −3.46404030276953560682936356407, −1.66428590858563208155672343236, 1.66056791280530419964166947111, 2.89596969529056212291170785903, 3.94194949488534628329746927292, 5.81827331240591844137998012270, 6.72619370226598606519558419993, 7.57108077637507915248482455837, 8.371165343615964896539927799642, 9.339027367221684243598958179816, 10.50930636070134091379359926344, 11.21017976426316485496170277792

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