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Properties

Label 2-380-76.31-c1-0-22
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: $\chi_{380} (31, \cdot )$
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

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

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