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Properties

Label 2-380-20.7-c1-0-53
Degree $2$
Conductor $380$
Sign $-0.999 - 0.0408i$
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  + (1.27 − 0.615i)2-s + (−1.93 − 1.93i)3-s + (1.24 − 1.56i)4-s + (−0.168 − 2.22i)5-s + (−3.66 − 1.27i)6-s + (−2.91 + 2.91i)7-s + (0.615 − 2.76i)8-s + 4.51i·9-s + (−1.58 − 2.73i)10-s − 3.32i·11-s + (−5.44 + 0.632i)12-s + (−0.313 + 0.313i)13-s + (−1.91 + 5.51i)14-s + (−3.99 + 4.64i)15-s + (−0.916 − 3.89i)16-s + (3.64 + 3.64i)17-s + ⋯
L(s)  = 1  + (0.900 − 0.435i)2-s + (−1.11 − 1.11i)3-s + (0.620 − 0.783i)4-s + (−0.0753 − 0.997i)5-s + (−1.49 − 0.520i)6-s + (−1.10 + 1.10i)7-s + (0.217 − 0.976i)8-s + 1.50i·9-s + (−0.501 − 0.864i)10-s − 1.00i·11-s + (−1.57 + 0.182i)12-s + (−0.0870 + 0.0870i)13-s + (−0.512 + 1.47i)14-s + (−1.03 + 1.20i)15-s + (−0.229 − 0.973i)16-s + (0.885 + 0.885i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.999 - 0.0408i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.999 - 0.0408i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0239314 + 1.17112i\)
\(L(\frac12)\) \(\approx\) \(0.0239314 + 1.17112i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.27 + 0.615i)T \)
5 \( 1 + (0.168 + 2.22i)T \)
19 \( 1 - T \)
good3 \( 1 + (1.93 + 1.93i)T + 3iT^{2} \)
7 \( 1 + (2.91 - 2.91i)T - 7iT^{2} \)
11 \( 1 + 3.32iT - 11T^{2} \)
13 \( 1 + (0.313 - 0.313i)T - 13iT^{2} \)
17 \( 1 + (-3.64 - 3.64i)T + 17iT^{2} \)
23 \( 1 + (2.24 + 2.24i)T + 23iT^{2} \)
29 \( 1 + 8.34iT - 29T^{2} \)
31 \( 1 + 0.286iT - 31T^{2} \)
37 \( 1 + (6.20 + 6.20i)T + 37iT^{2} \)
41 \( 1 - 4.08T + 41T^{2} \)
43 \( 1 + (-1.71 - 1.71i)T + 43iT^{2} \)
47 \( 1 + (-8.21 + 8.21i)T - 47iT^{2} \)
53 \( 1 + (-0.195 + 0.195i)T - 53iT^{2} \)
59 \( 1 - 1.88T + 59T^{2} \)
61 \( 1 - 6.98T + 61T^{2} \)
67 \( 1 + (0.844 - 0.844i)T - 67iT^{2} \)
71 \( 1 + 6.69iT - 71T^{2} \)
73 \( 1 + (6.74 - 6.74i)T - 73iT^{2} \)
79 \( 1 - 16.9T + 79T^{2} \)
83 \( 1 + (-4.69 - 4.69i)T + 83iT^{2} \)
89 \( 1 - 9.12iT - 89T^{2} \)
97 \( 1 + (5.27 + 5.27i)T + 97iT^{2} \)
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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

−11.39739154651448346409759680879, −10.26207557126649030650613666671, −9.146157741632435025693257569934, −7.920100829979756109189858321300, −6.56172209600539216909881767647, −5.77265205108392419102863693475, −5.49417416226734597584190747189, −3.80596769747081215639856282317, −2.21157019840513003253519826923, −0.65955387573065397207063842918, 3.14650739075609665515187182497, 3.93478809003886772992011012624, 4.96557555458740728630928044522, 5.97460844595562992955175228725, 6.94305849380053148622503682562, 7.44273900883178646821386364447, 9.545241713105854617299680387039, 10.25594185153039068330782268294, 10.83868620466981500556979273948, 11.83220518968449951186595184702

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