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Properties

Label 2-380-95.33-c1-0-4
Degree $2$
Conductor $380$
Sign $0.599 - 0.800i$
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  + (1.60 + 2.29i)3-s + (1.74 − 1.40i)5-s + (−0.546 − 0.146i)7-s + (−1.66 + 4.56i)9-s + (0.973 − 1.68i)11-s + (4.44 + 3.11i)13-s + (6.02 + 1.74i)15-s + (0.656 − 1.40i)17-s + (−4.30 + 0.693i)19-s + (−0.542 − 1.48i)21-s + (−8.31 − 0.727i)23-s + (1.06 − 4.88i)25-s + (−5.03 + 1.34i)27-s + (1.88 + 0.685i)29-s + (−6.30 + 3.63i)31-s + ⋯
L(s)  = 1  + (0.928 + 1.32i)3-s + (0.778 − 0.627i)5-s + (−0.206 − 0.0553i)7-s + (−0.553 + 1.52i)9-s + (0.293 − 0.508i)11-s + (1.23 + 0.863i)13-s + (1.55 + 0.450i)15-s + (0.159 − 0.341i)17-s + (−0.987 + 0.159i)19-s + (−0.118 − 0.325i)21-s + (−1.73 − 0.151i)23-s + (0.212 − 0.977i)25-s + (−0.968 + 0.259i)27-s + (0.349 + 0.127i)29-s + (−1.13 + 0.653i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.599 - 0.800i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.599 - 0.800i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79136 + 0.896802i\)
\(L(\frac12)\) \(\approx\) \(1.79136 + 0.896802i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.74 + 1.40i)T \)
19 \( 1 + (4.30 - 0.693i)T \)
good3 \( 1 + (-1.60 - 2.29i)T + (-1.02 + 2.81i)T^{2} \)
7 \( 1 + (0.546 + 0.146i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.973 + 1.68i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.44 - 3.11i)T + (4.44 + 12.2i)T^{2} \)
17 \( 1 + (-0.656 + 1.40i)T + (-10.9 - 13.0i)T^{2} \)
23 \( 1 + (8.31 + 0.727i)T + (22.6 + 3.99i)T^{2} \)
29 \( 1 + (-1.88 - 0.685i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (6.30 - 3.63i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.89 - 4.89i)T + 37iT^{2} \)
41 \( 1 + (5.71 + 1.00i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.926 + 10.5i)T + (-42.3 + 7.46i)T^{2} \)
47 \( 1 + (-5.99 + 2.79i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (-0.198 + 2.26i)T + (-52.1 - 9.20i)T^{2} \)
59 \( 1 + (-3.62 + 1.31i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-3.63 + 3.04i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-0.894 - 1.91i)T + (-43.0 + 51.3i)T^{2} \)
71 \( 1 + (-1.35 + 1.61i)T + (-12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.728 - 0.510i)T + (24.9 - 68.5i)T^{2} \)
79 \( 1 + (2.16 - 12.3i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-0.674 + 2.51i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (0.660 + 3.74i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (14.7 + 6.87i)T + (62.3 + 74.3i)T^{2} \)
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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

−11.23858584121183310218502970187, −10.25563612370244158453539325525, −9.675208770472621135434924330627, −8.589618846248445744601476394283, −8.538821791131978648807685127877, −6.57247803298057817294502614173, −5.53025871911160531324691355613, −4.30390656230350270187654570910, −3.56662949012887338707138533921, −2.00293531874005491820005409608, 1.59330590131657010279187568052, 2.59263963597519644600416786280, 3.80372036093114268401421558692, 5.94852594291329170227562173483, 6.41702529994974954293087290396, 7.55047833946341718650486911790, 8.271705589809583309677250005026, 9.257368047840675632184104737818, 10.22757608979955210471078991082, 11.23664393341334875680824041334

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