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Properties

Label 2-380-76.75-c1-0-0
Degree $2$
Conductor $380$
Sign $-0.896 + 0.443i$
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  + (0.482 + 1.32i)2-s − 0.428·3-s + (−1.53 + 1.28i)4-s − 5-s + (−0.206 − 0.569i)6-s + 1.38i·7-s + (−2.44 − 1.42i)8-s − 2.81·9-s + (−0.482 − 1.32i)10-s + 3.50i·11-s + (0.658 − 0.549i)12-s − 3.22i·13-s + (−1.83 + 0.666i)14-s + 0.428·15-s + (0.714 − 3.93i)16-s − 4.08·17-s + ⋯
L(s)  = 1  + (0.340 + 0.940i)2-s − 0.247·3-s + (−0.767 + 0.640i)4-s − 0.447·5-s + (−0.0843 − 0.232i)6-s + 0.522i·7-s + (−0.864 − 0.503i)8-s − 0.938·9-s + (−0.152 − 0.420i)10-s + 1.05i·11-s + (0.189 − 0.158i)12-s − 0.895i·13-s + (−0.491 + 0.178i)14-s + 0.110·15-s + (0.178 − 0.983i)16-s − 0.990·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.896 + 0.443i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.896 + 0.443i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.127453 - 0.545317i\)
\(L(\frac12)\) \(\approx\) \(0.127453 - 0.545317i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.482 - 1.32i)T \)
5 \( 1 + T \)
19 \( 1 + (4.23 + 1.02i)T \)
good3 \( 1 + 0.428T + 3T^{2} \)
7 \( 1 - 1.38iT - 7T^{2} \)
11 \( 1 - 3.50iT - 11T^{2} \)
13 \( 1 + 3.22iT - 13T^{2} \)
17 \( 1 + 4.08T + 17T^{2} \)
23 \( 1 - 7.75iT - 23T^{2} \)
29 \( 1 + 3.98iT - 29T^{2} \)
31 \( 1 + 0.741T + 31T^{2} \)
37 \( 1 - 8.37iT - 37T^{2} \)
41 \( 1 + 1.50iT - 41T^{2} \)
43 \( 1 - 3.59iT - 43T^{2} \)
47 \( 1 - 2.78iT - 47T^{2} \)
53 \( 1 - 7.46iT - 53T^{2} \)
59 \( 1 - 5.98T + 59T^{2} \)
61 \( 1 - 0.587T + 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 + 0.374T + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 + 15.9T + 79T^{2} \)
83 \( 1 - 10.2iT - 83T^{2} \)
89 \( 1 - 8.43iT - 89T^{2} \)
97 \( 1 + 4.99iT - 97T^{2} \)
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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

−11.98732221405440794078141527613, −11.19655426847427125986183962547, −9.857436759845147879167466569611, −8.838099167055639019101054243692, −8.097051153142108387996161513312, −7.12473097171711638723619679007, −6.09150148130155970011707879061, −5.21696605655437249445038738375, −4.18371675519265910596343218824, −2.76091602496750928369994385357, 0.33050831775407691144931936977, 2.35679079406651084891388788802, 3.69802951108085306383383396576, 4.58834211398060866141569188177, 5.84789059066089445964810824683, 6.80394000605389459193029548352, 8.595576937332779520375717965757, 8.782902956702266533518244652587, 10.34580914468263881716591241692, 11.02166258368999352499558714230

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