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Properties

Label 2-380-95.37-c1-0-9
Degree $2$
Conductor $380$
Sign $-0.264 + 0.964i$
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  + (1.63 − 1.52i)5-s + (−2.84 − 2.84i)7-s − 3i·9-s − 6.50·11-s + (1.69 + 1.69i)17-s + 4.35i·19-s + (6.35 − 6.35i)23-s + (0.362 − 4.98i)25-s + (−8.99 − 0.326i)35-s + (8.74 − 8.74i)43-s + (−4.56 − 4.91i)45-s + (5.35 + 5.35i)47-s + 9.20i·49-s + (−10.6 + 9.91i)55-s + 10.8·61-s + ⋯
L(s)  = 1  + (0.732 − 0.680i)5-s + (−1.07 − 1.07i)7-s − i·9-s − 1.96·11-s + (0.411 + 0.411i)17-s + 0.999i·19-s + (1.32 − 1.32i)23-s + (0.0725 − 0.997i)25-s + (−1.52 − 0.0552i)35-s + (1.33 − 1.33i)43-s + (−0.680 − 0.732i)45-s + (0.781 + 0.781i)47-s + 1.31i·49-s + (−1.43 + 1.33i)55-s + 1.38·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.264 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.264 + 0.964i$
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: \((2,\ 380,\ (\ :1/2),\ -0.264 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.653844 - 0.857696i\)
\(L(\frac12)\) \(\approx\) \(0.653844 - 0.857696i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 + 1.52i)T \)
19 \( 1 - 4.35iT \)
good3 \( 1 + 3iT^{2} \)
7 \( 1 + (2.84 + 2.84i)T + 7iT^{2} \)
11 \( 1 + 6.50T + 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (-1.69 - 1.69i)T + 17iT^{2} \)
23 \( 1 + (-6.35 + 6.35i)T - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-8.74 + 8.74i)T - 43iT^{2} \)
47 \( 1 + (-5.35 - 5.35i)T + 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-5.11 + 5.11i)T - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (3.64 - 3.64i)T - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97iT^{2} \)
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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

−10.66777042946918141737991908938, −10.24606052113809148004614954897, −9.402809523161800714076952501533, −8.370962545279839432620084704322, −7.25996401077697547118705022112, −6.24111124363604441206315034432, −5.31789327010472994082981907940, −3.99561280373648364577780159020, −2.74134400804949731630884967303, −0.69759881143613972003354907387, 2.49259648208351475528105512439, 2.93821372341480754828870362113, 5.21319324672348986888219649881, 5.60542842980191256983081276171, 6.92879070679215288206826172686, 7.78420542879521804463417181464, 9.065417071812718042849890043151, 9.826113548394936994341683706687, 10.65871008778158891529656865594, 11.40908414556028762188549693829

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