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Properties

Label 2-380-76.27-c1-0-32
Degree $2$
Conductor $380$
Sign $0.961 + 0.276i$
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  + (1.28 + 0.591i)2-s + (1.05 − 1.82i)3-s + (1.30 + 1.51i)4-s + (0.5 − 0.866i)5-s + (2.43 − 1.72i)6-s − 0.279i·7-s + (0.773 + 2.72i)8-s + (−0.719 − 1.24i)9-s + (1.15 − 0.817i)10-s − 2.67i·11-s + (4.14 − 0.773i)12-s + (−3.06 + 1.77i)13-s + (0.164 − 0.358i)14-s + (−1.05 − 1.82i)15-s + (−0.614 + 3.95i)16-s + (1.01 − 1.75i)17-s + ⋯
L(s)  = 1  + (0.908 + 0.418i)2-s + (0.608 − 1.05i)3-s + (0.650 + 0.759i)4-s + (0.223 − 0.387i)5-s + (0.992 − 0.702i)6-s − 0.105i·7-s + (0.273 + 0.961i)8-s + (−0.239 − 0.415i)9-s + (0.365 − 0.258i)10-s − 0.807i·11-s + (1.19 − 0.223i)12-s + (−0.851 + 0.491i)13-s + (0.0440 − 0.0957i)14-s + (−0.271 − 0.471i)15-s + (−0.153 + 0.988i)16-s + (0.246 − 0.426i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.961 + 0.276i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.961 + 0.276i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.70651 - 0.381249i\)
\(L(\frac12)\) \(\approx\) \(2.70651 - 0.381249i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 - 0.591i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (2.32 - 3.68i)T \)
good3 \( 1 + (-1.05 + 1.82i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 0.279iT - 7T^{2} \)
11 \( 1 + 2.67iT - 11T^{2} \)
13 \( 1 + (3.06 - 1.77i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.01 + 1.75i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.18 - 0.682i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.05 + 0.606i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.76T + 31T^{2} \)
37 \( 1 + 0.254iT - 37T^{2} \)
41 \( 1 + (-6.19 - 3.57i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.49 + 3.75i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.23 - 1.29i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.79 - 3.92i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.13 - 8.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.28 - 2.23i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.21 + 10.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.81 + 3.14i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.69 + 4.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.63 - 9.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.31iT - 83T^{2} \)
89 \( 1 + (-0.830 + 0.479i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.76 - 1.02i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.78878598843528664179305472636, −10.58619251327884334130910964072, −9.170540944091820689549454106940, −8.162037261931474361082748002551, −7.52178117626513342615547229740, −6.57151978808788886211344079330, −5.60481753696834483933676691395, −4.38547588465186057206280870203, −3.00015587067153623512303249195, −1.81804227635954946091840182084, 2.24598682641700977482730084324, 3.27892132529025411169722201036, 4.35744076984898473392933584844, 5.16378933735958012322300892463, 6.45570924880860386834720652790, 7.51314850107614143603045067224, 8.982217517969623314604435084209, 9.914947961680230165591635507047, 10.34038060684270116434039359768, 11.31619278428541612645097561668

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