Properties

Label 2-380-380.239-c2-0-92
Degree $2$
Conductor $380$
Sign $-0.196 + 0.980i$
Analytic cond. $10.3542$
Root an. cond. $3.21780$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.213 + 1.98i)2-s + (−0.0580 + 0.100i)3-s + (−3.90 + 0.849i)4-s + (2.96 − 4.02i)5-s + (−0.212 − 0.0939i)6-s − 7.81·7-s + (−2.52 − 7.59i)8-s + (4.49 + 7.78i)9-s + (8.63 + 5.03i)10-s + 3.02i·11-s + (0.141 − 0.442i)12-s + (−11.0 + 6.40i)13-s + (−1.67 − 15.5i)14-s + (0.232 + 0.531i)15-s + (14.5 − 6.64i)16-s + (−17.3 − 10.0i)17-s + ⋯
L(s)  = 1  + (0.106 + 0.994i)2-s + (−0.0193 + 0.0335i)3-s + (−0.977 + 0.212i)4-s + (0.593 − 0.805i)5-s + (−0.0353 − 0.0156i)6-s − 1.11·7-s + (−0.315 − 0.948i)8-s + (0.499 + 0.864i)9-s + (0.863 + 0.503i)10-s + 0.275i·11-s + (0.0117 − 0.0368i)12-s + (−0.853 + 0.492i)13-s + (−0.119 − 1.11i)14-s + (0.0155 + 0.0354i)15-s + (0.909 − 0.415i)16-s + (−1.01 − 0.588i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.196 + 0.980i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ -0.196 + 0.980i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.153418 - 0.187299i\)
\(L(\frac12)\) \(\approx\) \(0.153418 - 0.187299i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.213 - 1.98i)T \)
5 \( 1 + (-2.96 + 4.02i)T \)
19 \( 1 + (9.66 + 16.3i)T \)
good3 \( 1 + (0.0580 - 0.100i)T + (-4.5 - 7.79i)T^{2} \)
7 \( 1 + 7.81T + 49T^{2} \)
11 \( 1 - 3.02iT - 121T^{2} \)
13 \( 1 + (11.0 - 6.40i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (17.3 + 10.0i)T + (144.5 + 250. i)T^{2} \)
23 \( 1 + (17.7 + 30.6i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (20.2 + 35.1i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 - 31.8iT - 961T^{2} \)
37 \( 1 - 47.3iT - 1.36e3T^{2} \)
41 \( 1 + (-15.6 + 27.0i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-16.3 + 28.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-19.2 - 33.3i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (62.0 - 35.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (85.4 + 49.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-17.0 - 29.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (40.0 + 69.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-14.5 - 8.38i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-57.9 - 33.4i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-87.2 - 50.3i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 38.3T + 6.88e3T^{2} \)
89 \( 1 + (22.1 + 38.4i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (102. + 59.0i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.56150896093054458090193775866, −9.604465231528611415854433801130, −9.135061068703646746108853711123, −8.017796334789413014883123448250, −6.90225864425562638099121509050, −6.23610733745849082890422875059, −4.89657353667406705625674893400, −4.37668259596992910180426195308, −2.38369887740353308097493952371, −0.097614610503532437032830366776, 1.90582265099765445900817987202, 3.18052733459306465943123884043, 3.97898427693568948644124447013, 5.68785399425869637248091797169, 6.41701175237219558535887936095, 7.65757138200703983826064572963, 9.225111928688752653598040595088, 9.664185997941968061939973143546, 10.44709823257350443222024745211, 11.27128648282017741022374280204

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