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Properties

Label 2-380-380.379-c1-0-51
Degree $2$
Conductor $380$
Sign $-0.122 + 0.992i$
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  + (1.39 − 0.221i)2-s − 2.04i·3-s + (1.90 − 0.619i)4-s + (0.665 − 2.13i)5-s + (−0.453 − 2.85i)6-s − 2.07·7-s + (2.51 − 1.28i)8-s − 1.17·9-s + (0.456 − 3.12i)10-s + 4.08i·11-s + (−1.26 − 3.88i)12-s − 4.52·13-s + (−2.90 + 0.460i)14-s + (−4.36 − 1.36i)15-s + (3.23 − 2.35i)16-s + 6.58i·17-s + ⋯
L(s)  = 1  + (0.987 − 0.156i)2-s − 1.17i·3-s + (0.950 − 0.309i)4-s + (0.297 − 0.954i)5-s + (−0.184 − 1.16i)6-s − 0.785·7-s + (0.890 − 0.454i)8-s − 0.392·9-s + (0.144 − 0.989i)10-s + 1.23i·11-s + (−0.365 − 1.12i)12-s − 1.25·13-s + (−0.775 + 0.123i)14-s + (−1.12 − 0.351i)15-s + (0.808 − 0.588i)16-s + 1.59i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.122 + 0.992i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.122 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59511 - 1.80495i\)
\(L(\frac12)\) \(\approx\) \(1.59511 - 1.80495i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.221i)T \)
5 \( 1 + (-0.665 + 2.13i)T \)
19 \( 1 + (-4.31 - 0.590i)T \)
good3 \( 1 + 2.04iT - 3T^{2} \)
7 \( 1 + 2.07T + 7T^{2} \)
11 \( 1 - 4.08iT - 11T^{2} \)
13 \( 1 + 4.52T + 13T^{2} \)
17 \( 1 - 6.58iT - 17T^{2} \)
23 \( 1 - 7.09T + 23T^{2} \)
29 \( 1 + 5.90iT - 29T^{2} \)
31 \( 1 + 1.69T + 31T^{2} \)
37 \( 1 - 1.81T + 37T^{2} \)
41 \( 1 - 1.10iT - 41T^{2} \)
43 \( 1 - 7.73T + 43T^{2} \)
47 \( 1 + 9.98T + 47T^{2} \)
53 \( 1 - 2.18T + 53T^{2} \)
59 \( 1 + 9.39T + 59T^{2} \)
61 \( 1 - 4.95T + 61T^{2} \)
67 \( 1 - 13.3iT - 67T^{2} \)
71 \( 1 - 4.67T + 71T^{2} \)
73 \( 1 - 6.18iT - 73T^{2} \)
79 \( 1 + 4.04T + 79T^{2} \)
83 \( 1 + 7.46T + 83T^{2} \)
89 \( 1 - 0.553iT - 89T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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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

−11.64302552723702052770799780411, −10.13230547172991460490909031034, −9.512142083323084776354187744416, −7.915333997832085495640214377120, −7.15433034962940062767474912856, −6.30456340304103716379632337932, −5.24927074353721677306433379299, −4.19992400427360114672645526581, −2.52798093092974735794584871864, −1.41070139079176588871796032322, 2.99963726510501941773024337974, 3.23357381077206660638385525297, 4.80753075507507316549961053901, 5.52857319977800698221260333936, 6.78207938053724380337583779102, 7.42733623794248805744611104272, 9.234643419957944452638224261855, 9.847436776590652125558722795787, 10.90393409808437946362447381513, 11.35094171985234816732235633687

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