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Properties

Label 1-465-465.257-r0-0-0
Degree $1$
Conductor $465$
Sign $0.0768 + 0.997i$
Analytic cond. $2.15945$
Root an. cond. $2.15945$
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  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.743 + 0.669i)7-s + (0.951 + 0.309i)8-s + (0.978 + 0.207i)11-s + (−0.994 + 0.104i)13-s + (−0.978 + 0.207i)14-s + (−0.809 + 0.587i)16-s + (0.207 + 0.978i)17-s + (0.104 − 0.994i)19-s + (−0.743 + 0.669i)22-s + (0.951 + 0.309i)23-s + (0.5 − 0.866i)26-s + (0.406 − 0.913i)28-s + (−0.809 − 0.587i)29-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.743 + 0.669i)7-s + (0.951 + 0.309i)8-s + (0.978 + 0.207i)11-s + (−0.994 + 0.104i)13-s + (−0.978 + 0.207i)14-s + (−0.809 + 0.587i)16-s + (0.207 + 0.978i)17-s + (0.104 − 0.994i)19-s + (−0.743 + 0.669i)22-s + (0.951 + 0.309i)23-s + (0.5 − 0.866i)26-s + (0.406 − 0.913i)28-s + (−0.809 − 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0768 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0768 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.0768 + 0.997i$
Analytic conductor: \(2.15945\)
Root analytic conductor: \(2.15945\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 465,\ (0:\ ),\ 0.0768 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7735146410 + 0.7161595703i\)
\(L(\frac12)\) \(\approx\) \(0.7735146410 + 0.7161595703i\)
\(L(1)\) \(\approx\) \(0.7947125635 + 0.3863744349i\)
\(L(1)\) \(\approx\) \(0.7947125635 + 0.3863744349i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.587 + 0.809i)T \)
7 \( 1 + (0.743 + 0.669i)T \)
11 \( 1 + (0.978 + 0.207i)T \)
13 \( 1 + (-0.994 + 0.104i)T \)
17 \( 1 + (0.207 + 0.978i)T \)
19 \( 1 + (0.104 - 0.994i)T \)
23 \( 1 + (0.951 + 0.309i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (-0.913 + 0.406i)T \)
43 \( 1 + (0.994 + 0.104i)T \)
47 \( 1 + (0.587 + 0.809i)T \)
53 \( 1 + (-0.743 + 0.669i)T \)
59 \( 1 + (0.913 + 0.406i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (-0.207 + 0.978i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (0.406 + 0.913i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.951 + 0.309i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.61732812174510575429929255692, −22.51968971312614412026038945324, −21.98880923227795948902265123176, −20.74761201587122536805387613580, −20.440379642846167082840383082120, −19.41509232664706402936105387095, −18.67255875269339016684655190305, −17.695929029362494435300947479046, −16.94182711835261260304330425811, −16.40203132536059155203653941246, −14.762392844836307182753126619065, −14.10712643556632994308835479574, −13.05214025565850420774356151010, −11.99626693129144774832272042453, −11.40813990665567643599271754042, −10.421896816713143951130476807245, −9.5898910877490079236558377219, −8.67762919864732393178225470268, −7.62008140157829439552510128466, −6.948627398768704782825275798074, −5.20123672046209994458552776982, −4.22180845176736284543209208018, −3.22154212502422275641028054138, −1.93886727622013437793610966850, −0.85882894883945104226728184514, 1.24951712197598523294695540940, 2.38632191720797549374219898531, 4.22517408989108324835353089225, 5.13490358467897374183321698587, 6.088129528751954526485158649383, 7.12682366678524861476120068886, 7.94920176398476701452334672115, 9.02694890231827465606245742731, 9.52572383764283691183520913326, 10.82518473108718940828226280171, 11.64473922726974466551860310428, 12.7833177342427470315425426833, 14.00399143247386502486761211237, 14.96231781846283742211884056891, 15.14532520258130464551321381715, 16.520885103349716240090934550393, 17.3348044002150243790910438278, 17.76493492120807329350718418012, 19.03041024905650493100502259489, 19.44675927060998569774601820900, 20.55126024753120750013363193551, 21.78011099031864380210364671669, 22.36234552020483748009349237554, 23.5816584203360583230295846028, 24.25101887978682726466633395022

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