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Properties

Label 2-380-76.75-c1-0-23
Degree $2$
Conductor $380$
Sign $0.501 - 0.865i$
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  + (1.10 + 0.881i)2-s + 1.47·3-s + (0.444 + 1.94i)4-s + 5-s + (1.63 + 1.30i)6-s − 1.71i·7-s + (−1.22 + 2.54i)8-s − 0.818·9-s + (1.10 + 0.881i)10-s + 3.88i·11-s + (0.656 + 2.88i)12-s − 1.85i·13-s + (1.51 − 1.89i)14-s + 1.47·15-s + (−3.60 + 1.73i)16-s + 6.69·17-s + ⋯
L(s)  = 1  + (0.781 + 0.623i)2-s + 0.852·3-s + (0.222 + 0.974i)4-s + 0.447·5-s + (0.666 + 0.531i)6-s − 0.648i·7-s + (−0.434 + 0.900i)8-s − 0.272·9-s + (0.349 + 0.278i)10-s + 1.17i·11-s + (0.189 + 0.831i)12-s − 0.513i·13-s + (0.404 − 0.507i)14-s + 0.381·15-s + (−0.901 + 0.433i)16-s + 1.62·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.501 - 0.865i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.501 - 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31085 + 1.33139i\)
\(L(\frac12)\) \(\approx\) \(2.31085 + 1.33139i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.10 - 0.881i)T \)
5 \( 1 - T \)
19 \( 1 + (3.19 + 2.96i)T \)
good3 \( 1 - 1.47T + 3T^{2} \)
7 \( 1 + 1.71iT - 7T^{2} \)
11 \( 1 - 3.88iT - 11T^{2} \)
13 \( 1 + 1.85iT - 13T^{2} \)
17 \( 1 - 6.69T + 17T^{2} \)
23 \( 1 + 5.06iT - 23T^{2} \)
29 \( 1 + 4.80iT - 29T^{2} \)
31 \( 1 + 7.19T + 31T^{2} \)
37 \( 1 - 8.50iT - 37T^{2} \)
41 \( 1 - 3.02iT - 41T^{2} \)
43 \( 1 + 5.74iT - 43T^{2} \)
47 \( 1 - 5.37iT - 47T^{2} \)
53 \( 1 + 11.7iT - 53T^{2} \)
59 \( 1 + 5.83T + 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 + 6.04T + 67T^{2} \)
71 \( 1 - 1.61T + 71T^{2} \)
73 \( 1 + 13.3T + 73T^{2} \)
79 \( 1 - 4.57T + 79T^{2} \)
83 \( 1 - 12.5iT - 83T^{2} \)
89 \( 1 + 8.64iT - 89T^{2} \)
97 \( 1 - 15.1iT - 97T^{2} \)
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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

−11.77450409237880814869844864094, −10.50564802260713549597372871244, −9.581012517241043255866816845212, −8.461208853652420105409106333708, −7.70622574967963767392804345852, −6.83511025061056274814479503159, −5.67064156794236537718217667685, −4.57805766823106669089943317452, −3.43002048224189151249668663945, −2.31063487523113169290668174936, 1.75214653268319458061708291799, 2.99671192408189358757744125407, 3.76438462334652042307324908310, 5.56130919727790118368549371126, 5.86159161555867678517141720779, 7.47797797589479975590668018134, 8.763038788785750074499603698040, 9.303434578111047928151539966012, 10.40253422783251428897278386523, 11.30979715601844994618594350618

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