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Properties

Label 2-380-380.379-c1-0-21
Degree $2$
Conductor $380$
Sign $0.284 + 0.958i$
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  + (−0.203 − 1.39i)2-s + 1.12i·3-s + (−1.91 + 0.568i)4-s − 2.23·5-s + (1.57 − 0.227i)6-s + (1.18 + 2.56i)8-s + 1.74·9-s + (0.453 + 3.12i)10-s − 5.95i·11-s + (−0.637 − 2.15i)12-s + 6.51·13-s − 2.50i·15-s + (3.35 − 2.17i)16-s + (−0.353 − 2.43i)18-s − 4.35i·19-s + (4.28 − 1.27i)20-s + ⋯
L(s)  = 1  + (−0.143 − 0.989i)2-s + 0.647i·3-s + (−0.958 + 0.284i)4-s − 0.999·5-s + (0.641 − 0.0930i)6-s + (0.418 + 0.908i)8-s + 0.580·9-s + (0.143 + 0.989i)10-s − 1.79i·11-s + (−0.184 − 0.621i)12-s + 1.80·13-s − 0.647i·15-s + (0.838 − 0.544i)16-s + (−0.0832 − 0.574i)18-s − 0.999i·19-s + (0.958 − 0.284i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.284 + 0.958i$
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: \((2,\ 380,\ (\ :1/2),\ 0.284 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.836269 - 0.624385i\)
\(L(\frac12)\) \(\approx\) \(0.836269 - 0.624385i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.203 + 1.39i)T \)
5 \( 1 + 2.23T \)
19 \( 1 + 4.35iT \)
good3 \( 1 - 1.12iT - 3T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 5.95iT - 11T^{2} \)
13 \( 1 - 6.51T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 8.01T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 5.09T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 1.11T + 61T^{2} \)
67 \( 1 + 13.7iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−11.05834075811814045706146179729, −10.67565768786094845398836671144, −9.310558072298132654524540175916, −8.631813473856212505258583269451, −7.85945440204638616762641532278, −6.24957611882866798165079138030, −4.84351286270871015910408530282, −3.80780112196605458523585040753, −3.20927807345127265803627934958, −0.923751885223943354440011888904, 1.38777519347960803787656745516, 3.83884039823818018222714491154, 4.60562065818393571572910093184, 6.11486326529476646654984497767, 6.96982503180036210989343269237, 7.71289619431099041117017866738, 8.367191465049300895786259112127, 9.552662669023053945827538661577, 10.47062014236348868872866181926, 11.70965988934871158569501687148

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