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Properties

Label 2-380-20.3-c1-0-30
Degree $2$
Conductor $380$
Sign $0.991 - 0.128i$
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  + (1.22 − 0.707i)2-s + (−1.27 + 1.27i)3-s + (0.999 − 1.73i)4-s + (2.08 + 0.796i)5-s + (−0.658 + 2.45i)6-s + (0.691 + 0.691i)7-s + (−8.52e−5 − 2.82i)8-s − 0.236i·9-s + (3.12 − 0.501i)10-s + 2.48i·11-s + (0.931 + 3.47i)12-s + (0.299 + 0.299i)13-s + (1.33 + 0.357i)14-s + (−3.67 + 1.64i)15-s + (−2.00 − 3.46i)16-s + (1.98 − 1.98i)17-s + ⋯
L(s)  = 1  + (0.866 − 0.500i)2-s + (−0.734 + 0.734i)3-s + (0.499 − 0.866i)4-s + (0.934 + 0.356i)5-s + (−0.268 + 1.00i)6-s + (0.261 + 0.261i)7-s + (−3.01e−5 − 0.999i)8-s − 0.0789i·9-s + (0.987 − 0.158i)10-s + 0.748i·11-s + (0.268 + 1.00i)12-s + (0.0830 + 0.0830i)13-s + (0.356 + 0.0956i)14-s + (−0.947 + 0.424i)15-s + (−0.500 − 0.866i)16-s + (0.482 − 0.482i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.991 - 0.128i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.991 - 0.128i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07378 + 0.134280i\)
\(L(\frac12)\) \(\approx\) \(2.07378 + 0.134280i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
5 \( 1 + (-2.08 - 0.796i)T \)
19 \( 1 - T \)
good3 \( 1 + (1.27 - 1.27i)T - 3iT^{2} \)
7 \( 1 + (-0.691 - 0.691i)T + 7iT^{2} \)
11 \( 1 - 2.48iT - 11T^{2} \)
13 \( 1 + (-0.299 - 0.299i)T + 13iT^{2} \)
17 \( 1 + (-1.98 + 1.98i)T - 17iT^{2} \)
23 \( 1 + (-3.56 + 3.56i)T - 23iT^{2} \)
29 \( 1 - 5.50iT - 29T^{2} \)
31 \( 1 - 1.55iT - 31T^{2} \)
37 \( 1 + (-1.57 + 1.57i)T - 37iT^{2} \)
41 \( 1 + 8.98T + 41T^{2} \)
43 \( 1 + (-0.612 + 0.612i)T - 43iT^{2} \)
47 \( 1 + (7.25 + 7.25i)T + 47iT^{2} \)
53 \( 1 + (-0.698 - 0.698i)T + 53iT^{2} \)
59 \( 1 + 6.30T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 + (9.08 + 9.08i)T + 67iT^{2} \)
71 \( 1 - 7.70iT - 71T^{2} \)
73 \( 1 + (1.79 + 1.79i)T + 73iT^{2} \)
79 \( 1 - 0.823T + 79T^{2} \)
83 \( 1 + (-0.768 + 0.768i)T - 83iT^{2} \)
89 \( 1 + 10.3iT - 89T^{2} \)
97 \( 1 + (5.43 - 5.43i)T - 97iT^{2} \)
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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

−11.34371929832973671560582353893, −10.47819328407576586728459262322, −10.03826589746805402186634178872, −9.042986288330194201039953739396, −7.21347999648618167624995391938, −6.24001062725099981219776515554, −5.19798373806770462847902684876, −4.77654901017572934141736641118, −3.19902569442130128593347679498, −1.84822515176138973980845249938, 1.45983130749709916125100908187, 3.20030155742491726192753925124, 4.72867541001664992429176524721, 5.80713331842053639282698197693, 6.19526413716310244699661135504, 7.30299053534543396414334593430, 8.264587517069716361633385770160, 9.438599589526190902725089477003, 10.77374922358498506346749629389, 11.59820463466114368265583544394

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