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Properties

Label 2-460-115.68-c1-0-7
Degree $2$
Conductor $460$
Sign $-0.631 + 0.775i$
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  + (−1 − i)3-s + (−1.83 + 1.28i)5-s + (1.83 + 1.83i)7-s − i·9-s − 3.66i·11-s + (−3.70 − 3.70i)13-s + (3.11 + 0.546i)15-s + (0.737 + 0.737i)17-s − 4.75·19-s − 3.66i·21-s + (3.23 − 3.53i)23-s + (1.70 − 4.70i)25-s + (−4 + 4i)27-s − 8.70i·29-s − 10.1·31-s + ⋯
L(s)  = 1  + (−0.577 − 0.577i)3-s + (−0.818 + 0.574i)5-s + (0.691 + 0.691i)7-s − 0.333i·9-s − 1.10i·11-s + (−1.02 − 1.02i)13-s + (0.804 + 0.141i)15-s + (0.178 + 0.178i)17-s − 1.09·19-s − 0.798i·21-s + (0.675 − 0.737i)23-s + (0.340 − 0.940i)25-s + (−0.769 + 0.769i)27-s − 1.61i·29-s − 1.81·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.631 + 0.775i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ -0.631 + 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.260194 - 0.547272i\)
\(L(\frac12)\) \(\approx\) \(0.260194 - 0.547272i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.83 - 1.28i)T \)
23 \( 1 + (-3.23 + 3.53i)T \)
good3 \( 1 + (1 + i)T + 3iT^{2} \)
7 \( 1 + (-1.83 - 1.83i)T + 7iT^{2} \)
11 \( 1 + 3.66iT - 11T^{2} \)
13 \( 1 + (3.70 + 3.70i)T + 13iT^{2} \)
17 \( 1 + (-0.737 - 0.737i)T + 17iT^{2} \)
19 \( 1 + 4.75T + 19T^{2} \)
29 \( 1 + 8.70iT - 29T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 + (-0.737 - 0.737i)T + 37iT^{2} \)
41 \( 1 - 1.29T + 41T^{2} \)
43 \( 1 + (7.86 - 7.86i)T - 43iT^{2} \)
47 \( 1 + (-6.40 + 6.40i)T - 47iT^{2} \)
53 \( 1 + (-2.92 + 2.92i)T - 53iT^{2} \)
59 \( 1 - 2.70iT - 59T^{2} \)
61 \( 1 + 12.0iT - 61T^{2} \)
67 \( 1 + (-6.58 - 6.58i)T + 67iT^{2} \)
71 \( 1 - 8.70T + 71T^{2} \)
73 \( 1 + (-10.4 - 10.4i)T + 73iT^{2} \)
79 \( 1 + 7.70T + 79T^{2} \)
83 \( 1 + (1.83 - 1.83i)T - 83iT^{2} \)
89 \( 1 + 13.9T + 89T^{2} \)
97 \( 1 + (8.25 + 8.25i)T + 97iT^{2} \)
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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

−11.15890531794671064598544632368, −10.01914167293652003443207882173, −8.632830132865555282234057328113, −8.005504103013861042050495616560, −7.01040257485143375438847405051, −6.07256787362247151678225396080, −5.17144310553677191655877505877, −3.72954674058311345377692518204, −2.45600868455045993838182632356, −0.39547779081891983190936324284, 1.82694397372285427955597590784, 3.94056031097119556921906432811, 4.69040410905823436668684182030, 5.24695830218369977346099610907, 7.13086285424879354113125631991, 7.47432286625651642585900932725, 8.768003325863919550328434473434, 9.638179312228485668753942897884, 10.72439538392217789635922924023, 11.19785620491655282272866000852

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