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Properties

Label 2-380-380.379-c1-0-54
Degree $2$
Conductor $380$
Sign $-0.284 - 0.958i$
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  + (−1.39 + 0.203i)2-s − 3.27i·3-s + (1.91 − 0.568i)4-s − 2.23·5-s + (0.665 + 4.58i)6-s + (−2.56 + 1.18i)8-s − 7.74·9-s + (3.12 − 0.453i)10-s + 5.95i·11-s + (−1.86 − 6.28i)12-s − 3.08·13-s + 7.32i·15-s + (3.35 − 2.17i)16-s + (10.8 − 1.57i)18-s − 4.35i·19-s + (−4.28 + 1.27i)20-s + ⋯
L(s)  = 1  + (−0.989 + 0.143i)2-s − 1.89i·3-s + (0.958 − 0.284i)4-s − 0.999·5-s + (0.271 + 1.87i)6-s + (−0.908 + 0.418i)8-s − 2.58·9-s + (0.989 − 0.143i)10-s + 1.79i·11-s + (−0.537 − 1.81i)12-s − 0.856·13-s + 1.89i·15-s + (0.838 − 0.544i)16-s + (2.55 − 0.370i)18-s − 0.999i·19-s + (−0.958 + 0.284i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.284 - 0.958i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.284 - 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0104008 + 0.0139303i\)
\(L(\frac12)\) \(\approx\) \(0.0104008 + 0.0139303i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.203i)T \)
5 \( 1 + 2.23T \)
19 \( 1 + 4.35iT \)
good3 \( 1 + 3.27iT - 3T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 5.95iT - 11T^{2} \)
13 \( 1 + 3.08T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 9.15T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 13.6T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 1.11T + 61T^{2} \)
67 \( 1 - 8.95iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 15.8T + 97T^{2} \)
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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

−10.89860245580978020848164254761, −9.580992462916081582394052827360, −8.540198076354653695281460223164, −7.66959149300912268912832310523, −7.16033243627309237808848181319, −6.61705219046248958657928536773, −5.01999429958534778549308450456, −2.79304424037988609126098550892, −1.68776458509567614087402177296, −0.01523713701224084538483115035, 3.10170493374417015022880739045, 3.69256321710576632179851837789, 5.07548344739222240135139277149, 6.23372897818281593805209667282, 7.86371073533781396770935333318, 8.520630309515220033761514743673, 9.279521804500518025557892390561, 10.22608407820398925021879269357, 10.90014593016301721115418633205, 11.47627150473878713989364177989

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