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Properties

Label 2-460-115.22-c1-0-9
Degree $2$
Conductor $460$
Sign $0.667 + 0.744i$
Analytic cond. $3.67311$
Root an. cond. $1.91653$
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  + (2.03 − 2.03i)3-s + (1.06 + 1.96i)5-s + (−0.641 + 0.641i)7-s − 5.28i·9-s − 3.68i·11-s + (1.51 − 1.51i)13-s + (6.17 + 1.82i)15-s + (−0.140 + 0.140i)17-s + 1.03·19-s + 2.61i·21-s + (4.55 + 1.51i)23-s + (−2.71 + 4.19i)25-s + (−4.66 − 4.66i)27-s + 4.36i·29-s + 3.07·31-s + ⋯
L(s)  = 1  + (1.17 − 1.17i)3-s + (0.477 + 0.878i)5-s + (−0.242 + 0.242i)7-s − 1.76i·9-s − 1.11i·11-s + (0.420 − 0.420i)13-s + (1.59 + 0.470i)15-s + (−0.0341 + 0.0341i)17-s + 0.237·19-s + 0.569i·21-s + (0.948 + 0.316i)23-s + (−0.543 + 0.839i)25-s + (−0.896 − 0.896i)27-s + 0.809i·29-s + 0.551·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.667 + 0.744i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.667 + 0.744i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96469 - 0.877964i\)
\(L(\frac12)\) \(\approx\) \(1.96469 - 0.877964i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.06 - 1.96i)T \)
23 \( 1 + (-4.55 - 1.51i)T \)
good3 \( 1 + (-2.03 + 2.03i)T - 3iT^{2} \)
7 \( 1 + (0.641 - 0.641i)T - 7iT^{2} \)
11 \( 1 + 3.68iT - 11T^{2} \)
13 \( 1 + (-1.51 + 1.51i)T - 13iT^{2} \)
17 \( 1 + (0.140 - 0.140i)T - 17iT^{2} \)
19 \( 1 - 1.03T + 19T^{2} \)
29 \( 1 - 4.36iT - 29T^{2} \)
31 \( 1 - 3.07T + 31T^{2} \)
37 \( 1 + (3.28 - 3.28i)T - 37iT^{2} \)
41 \( 1 + 8.10T + 41T^{2} \)
43 \( 1 + (4.71 + 4.71i)T + 43iT^{2} \)
47 \( 1 + (1.77 + 1.77i)T + 47iT^{2} \)
53 \( 1 + (10.0 + 10.0i)T + 53iT^{2} \)
59 \( 1 - 6.32iT - 59T^{2} \)
61 \( 1 - 13.9iT - 61T^{2} \)
67 \( 1 + (7.97 - 7.97i)T - 67iT^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 + (-0.0704 + 0.0704i)T - 73iT^{2} \)
79 \( 1 + 9.66T + 79T^{2} \)
83 \( 1 + (-5.33 - 5.33i)T + 83iT^{2} \)
89 \( 1 + 8.40T + 89T^{2} \)
97 \( 1 + (-7.74 + 7.74i)T - 97iT^{2} \)
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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

−10.95974792745547949949649134721, −9.938436979562203865878056815506, −8.846706793009535899254739765340, −8.300930829414460324990960469751, −7.19234547436621051910544015046, −6.56326635045712280730174299396, −5.55029817292181350430094772019, −3.34194238982329110591060197563, −2.90281329147586864125236635955, −1.47895472777557611901324905228, 1.91867575213781510551878697250, 3.28987659428739480150730304438, 4.43059168404742293633690625397, 5.00405745735491732132630697582, 6.55179727990031173903925161478, 7.87973974650985775006448188518, 8.693625342386973228098115775160, 9.516302117446138897590196096087, 9.860369636090636061696638868530, 10.86847290624587899797869717237

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