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Properties

Label 2-380-5.4-c1-0-2
Degree $2$
Conductor $380$
Sign $0.860 - 0.509i$
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  − 1.62i·3-s + (−1.92 + 1.13i)5-s + 4.74i·7-s + 0.367·9-s + 4.48·11-s − 0.843i·13-s + (1.84 + 3.12i)15-s + 5.52i·17-s − 19-s + 7.69·21-s + 0.779i·23-s + (2.40 − 4.38i)25-s − 5.46i·27-s + 10.6·29-s − 8.65·31-s + ⋯
L(s)  = 1  − 0.936i·3-s + (−0.860 + 0.509i)5-s + 1.79i·7-s + 0.122·9-s + 1.35·11-s − 0.233i·13-s + (0.477 + 0.805i)15-s + 1.33i·17-s − 0.229·19-s + 1.67·21-s + 0.162i·23-s + (0.480 − 0.876i)25-s − 1.05i·27-s + 1.97·29-s − 1.55·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.860 - 0.509i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.860 - 0.509i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18906 + 0.325611i\)
\(L(\frac12)\) \(\approx\) \(1.18906 + 0.325611i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.92 - 1.13i)T \)
19 \( 1 + T \)
good3 \( 1 + 1.62iT - 3T^{2} \)
7 \( 1 - 4.74iT - 7T^{2} \)
11 \( 1 - 4.48T + 11T^{2} \)
13 \( 1 + 0.843iT - 13T^{2} \)
17 \( 1 - 5.52iT - 17T^{2} \)
23 \( 1 - 0.779iT - 23T^{2} \)
29 \( 1 - 10.6T + 29T^{2} \)
31 \( 1 + 8.65T + 31T^{2} \)
37 \( 1 - 1.62iT - 37T^{2} \)
41 \( 1 - 4.73T + 41T^{2} \)
43 \( 1 - 9.67iT - 43T^{2} \)
47 \( 1 - 3.18iT - 47T^{2} \)
53 \( 1 - 6.17iT - 53T^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 - 6.48T + 61T^{2} \)
67 \( 1 + 14.8iT - 67T^{2} \)
71 \( 1 - 0.303T + 71T^{2} \)
73 \( 1 + 10.0iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 0.779iT - 83T^{2} \)
89 \( 1 - 5.69T + 89T^{2} \)
97 \( 1 + 6.17iT - 97T^{2} \)
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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

−11.71333612030390045997778491679, −10.76078427801281482124902028121, −9.388844744677601364008635013234, −8.503034865093834700205616295596, −7.76529547791110415553982635775, −6.53440431202834257376805853023, −6.07507273122625144760597227590, −4.40588999962048822561800538431, −3.04310294824650483082339864215, −1.68638375769555239659888585623, 0.943065202752462097390512680432, 3.63011310106241699221720626587, 4.16779928803524231732748868621, 4.93776158098183182670679757316, 6.85423798645452880874925908282, 7.35056746237203191274917105003, 8.687500721243930878836313458371, 9.515224857435619703357113035672, 10.37776766041501454665970790922, 11.17227734270712792089901274290

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