Properties

Label 2-380-20.7-c1-0-49
Degree $2$
Conductor $380$
Sign $0.0898 + 0.995i$
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 − i)2-s + (1 + i)3-s − 2i·4-s + (−2 − i)5-s + 2·6-s + (2 − 2i)7-s + (−2 − 2i)8-s i·9-s + (−3 + i)10-s + (2 − 2i)12-s − 4i·14-s + (−1 − 3i)15-s − 4·16-s + (5 + 5i)17-s + (−1 − i)18-s + 19-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (0.577 + 0.577i)3-s i·4-s + (−0.894 − 0.447i)5-s + 0.816·6-s + (0.755 − 0.755i)7-s + (−0.707 − 0.707i)8-s − 0.333i·9-s + (−0.948 + 0.316i)10-s + (0.577 − 0.577i)12-s − 1.06i·14-s + (−0.258 − 0.774i)15-s − 16-s + (1.21 + 1.21i)17-s + (−0.235 − 0.235i)18-s + 0.229·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0898 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0898 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.0898 + 0.995i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.0898 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55084 - 1.41729i\)
\(L(\frac12)\) \(\approx\) \(1.55084 - 1.41729i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
5 \( 1 + (2 + i)T \)
19 \( 1 - T \)
good3 \( 1 + (-1 - i)T + 3iT^{2} \)
7 \( 1 + (-2 + 2i)T - 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (-5 - 5i)T + 17iT^{2} \)
23 \( 1 + (4 + 4i)T + 23iT^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + (-6 - 6i)T + 43iT^{2} \)
47 \( 1 + (-2 + 2i)T - 47iT^{2} \)
53 \( 1 + (10 - 10i)T - 53iT^{2} \)
59 \( 1 + 10T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + (3 - 3i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-5 + 5i)T - 73iT^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + (4 + 4i)T + 83iT^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + (10 + 10i)T + 97iT^{2} \)
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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

−11.06286219608012253202174852197, −10.46692530324337044241769577694, −9.466364235530592190458860281079, −8.459582899911114216542190792991, −7.55959236752472271406399563547, −6.07806367325204372861851192403, −4.70908441683338009113550475769, −4.04665746341830002899512634653, −3.20506358796431622833557285978, −1.23780297017531345497225522510, 2.36203056334514511723485276137, 3.43713756132659844980657979210, 4.77815000529713457625027552916, 5.73999413700006855073118716641, 7.09909375133523476221715656395, 7.83868937167427018725984537670, 8.191806405495216471193949116886, 9.445663812620590631911760050741, 11.06583849387050060505520002028, 11.87636244189049786114796173309

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