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Properties

Label 2-465-31.25-c1-0-10
Degree $2$
Conductor $465$
Sign $0.987 + 0.159i$
Analytic cond. $3.71304$
Root an. cond. $1.92692$
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  + (0.5 + 0.866i)3-s − 2·4-s + (0.5 − 0.866i)5-s + (−0.499 + 0.866i)9-s + (2.71 − 4.69i)11-s + (−1 − 1.73i)12-s + (2.21 − 3.83i)13-s + 0.999·15-s + 4·16-s + (2 + 3.46i)17-s + (4.21 + 7.29i)19-s + (−1 + 1.73i)20-s + 2·23-s + (−0.499 − 0.866i)25-s − 0.999·27-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s − 4-s + (0.223 − 0.387i)5-s + (−0.166 + 0.288i)9-s + (0.817 − 1.41i)11-s + (−0.288 − 0.499i)12-s + (0.613 − 1.06i)13-s + 0.258·15-s + 16-s + (0.485 + 0.840i)17-s + (0.966 + 1.67i)19-s + (−0.223 + 0.387i)20-s + 0.417·23-s + (−0.0999 − 0.173i)25-s − 0.192·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.987 + 0.159i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ 0.987 + 0.159i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37381 - 0.110553i\)
\(L(\frac12)\) \(\approx\) \(1.37381 - 0.110553i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (4.71 + 2.96i)T \)
good2 \( 1 + 2T^{2} \)
7 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.71 + 4.69i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.21 + 3.83i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.21 - 7.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 - 3.42T + 29T^{2} \)
37 \( 1 + (4.21 + 7.29i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.712 + 1.23i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.21 + 5.56i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.84T + 47T^{2} \)
53 \( 1 + (5.42 - 9.39i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.71 - 4.69i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 4.57T + 61T^{2} \)
67 \( 1 + (3.42 - 5.93i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.71 + 6.42i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.787 + 1.36i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.13 + 10.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3 - 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
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

−10.71876887906711054259191410018, −10.11345494855088726186789012050, −9.045812652557284371545611368954, −8.559876277466067173873015632214, −7.74610402805400982243928789770, −5.77817723870296404482042259802, −5.56253902827618540760329984904, −3.91395765428257144907967478924, −3.44063337915542185001376725258, −1.08665027297194875202601513417, 1.37764538744162485237260439653, 3.00725726889165323593852849790, 4.30918016200989616783784133325, 5.20884421801332166326545654080, 6.74653510157801272387340295005, 7.18779109108112101095399144720, 8.518961527589857487916026550754, 9.399667510177592611902232299626, 9.731934925439894812575361938068, 11.19730912372365259803187121286

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