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Properties

Label 2-465-465.299-c0-0-1
Degree $2$
Conductor $465$
Sign $0.957 + 0.289i$
Analytic cond. $0.232065$
Root an. cond. $0.481731$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0646 − 0.198i)2-s + (0.669 + 0.743i)3-s + (0.773 − 0.562i)4-s + (−0.5 − 0.866i)5-s + (0.104 − 0.181i)6-s + (−0.330 − 0.240i)8-s + (−0.104 + 0.994i)9-s + (−0.139 + 0.155i)10-s + (0.935 + 0.198i)12-s + (0.309 − 0.951i)15-s + (0.269 − 0.828i)16-s + (−0.913 + 0.406i)17-s + (0.204 − 0.0434i)18-s + (−1.30 − 0.278i)19-s + (−0.873 − 0.388i)20-s + ⋯
L(s)  = 1  + (−0.0646 − 0.198i)2-s + (0.669 + 0.743i)3-s + (0.773 − 0.562i)4-s + (−0.5 − 0.866i)5-s + (0.104 − 0.181i)6-s + (−0.330 − 0.240i)8-s + (−0.104 + 0.994i)9-s + (−0.139 + 0.155i)10-s + (0.935 + 0.198i)12-s + (0.309 − 0.951i)15-s + (0.269 − 0.828i)16-s + (−0.913 + 0.406i)17-s + (0.204 − 0.0434i)18-s + (−1.30 − 0.278i)19-s + (−0.873 − 0.388i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.957 + 0.289i$
Analytic conductor: \(0.232065\)
Root analytic conductor: \(0.481731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :0),\ 0.957 + 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.063511905\)
\(L(\frac12)\) \(\approx\) \(1.063511905\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.669 - 0.743i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (0.0646 + 0.198i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (0.978 - 0.207i)T^{2} \)
11 \( 1 + (-0.669 - 0.743i)T^{2} \)
13 \( 1 + (-0.913 + 0.406i)T^{2} \)
17 \( 1 + (0.913 - 0.406i)T + (0.669 - 0.743i)T^{2} \)
19 \( 1 + (1.30 + 0.278i)T + (0.913 + 0.406i)T^{2} \)
23 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.104 + 0.994i)T^{2} \)
43 \( 1 + (-0.913 - 0.406i)T^{2} \)
47 \( 1 + (-0.564 + 1.73i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.0646 - 0.614i)T + (-0.978 - 0.207i)T^{2} \)
59 \( 1 + (0.104 - 0.994i)T^{2} \)
61 \( 1 + 1.95T + T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.978 + 0.207i)T^{2} \)
73 \( 1 + (-0.669 - 0.743i)T^{2} \)
79 \( 1 + (-1.22 + 0.544i)T + (0.669 - 0.743i)T^{2} \)
83 \( 1 + (-1.22 + 1.35i)T + (-0.104 - 0.994i)T^{2} \)
89 \( 1 + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−10.98043439068602197330804480203, −10.48528460062129447368261672002, −9.276997609376890293244099627090, −8.789609386162726055124259637021, −7.72959444595804574929002119412, −6.63048662409717983142220910995, −5.30487194165589563885025111805, −4.43150606191911858825752555521, −3.20449759616214397780605774626, −1.83174706930767172794147089292, 2.25741699819250812528410686858, 3.02517143125482888083573226766, 4.23141662749007369210737394177, 6.34361309325417925790390725140, 6.69837174847486490258235836325, 7.65169289179704128008766330888, 8.295157817779525918881143930301, 9.247045365842631134471111647921, 10.72467317280276399724322834487, 11.25338508136275490974927765214

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