Properties

Label 2-465-465.464-c1-0-17
Degree $2$
Conductor $465$
Sign $0.189 - 0.981i$
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  + (1.70 + 0.328i)3-s − 2·4-s + 2.23i·5-s + (2.78 + 1.11i)9-s + (−3.40 − 0.657i)12-s + 1.93·13-s + (−0.735 + 3.80i)15-s + 4·16-s + 6.29i·17-s − 5.56·19-s − 4.47i·20-s + 8.26i·23-s − 5.00·25-s + (4.36 + 2.81i)27-s + 5.56·31-s + ⋯
L(s)  = 1  + (0.981 + 0.189i)3-s − 4-s + 0.999i·5-s + (0.927 + 0.372i)9-s + (−0.981 − 0.189i)12-s + 0.535·13-s + (−0.189 + 0.981i)15-s + 16-s + 1.52i·17-s − 1.27·19-s − 0.999i·20-s + 1.72i·23-s − 1.00·25-s + (0.840 + 0.542i)27-s + 1.00·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.189 - 0.981i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (464, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ 0.189 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16605 + 0.962241i\)
\(L(\frac12)\) \(\approx\) \(1.16605 + 0.962241i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 - 0.328i)T \)
5 \( 1 - 2.23iT \)
31 \( 1 - 5.56T \)
good2 \( 1 + 2T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 1.93T + 13T^{2} \)
17 \( 1 - 6.29iT - 17T^{2} \)
19 \( 1 + 5.56T + 19T^{2} \)
23 \( 1 - 8.26iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
37 \( 1 + 8.27T + 37T^{2} \)
41 \( 1 + 12.4iT - 41T^{2} \)
43 \( 1 - 12.1T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 14.5iT - 53T^{2} \)
59 \( 1 + 12.4iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 2.23iT - 71T^{2} \)
73 \( 1 - 6.34T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 10.2iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.83308129639883588000546878775, −10.31916734225104531427203946434, −9.387837057440030740673914866786, −8.527485631195517174667933400447, −7.86940469608264038254430055491, −6.71551255368042693379431860259, −5.53466754508751918167463270201, −3.99750305951344250513853397113, −3.57189996869489518979486013466, −1.98513692360736737839634951075, 0.926444403510182446851935432413, 2.69081857881804697767494012632, 4.20705322172005255109898471833, 4.68947915099540651696401274988, 6.12246667238778666260825432031, 7.47274442869828479589962627025, 8.531373779300826397354613747785, 8.794897297409404493644999976354, 9.647160985462610107112905537217, 10.57922642408553514668335305140

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