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Properties

Label 1-465-465.194-r1-0-0
Degree $1$
Conductor $465$
Sign $-0.962 + 0.272i$
Analytic cond. $49.9711$
Root an. cond. $49.9711$
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  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.309 − 0.951i)11-s + (0.809 − 0.587i)13-s + (−0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)17-s + (−0.809 − 0.587i)19-s + (−0.309 + 0.951i)22-s + (0.309 − 0.951i)23-s − 26-s + (0.809 − 0.587i)28-s + (0.809 + 0.587i)29-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.309 − 0.951i)11-s + (0.809 − 0.587i)13-s + (−0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)17-s + (−0.809 − 0.587i)19-s + (−0.309 + 0.951i)22-s + (0.309 − 0.951i)23-s − 26-s + (0.809 − 0.587i)28-s + (0.809 + 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.962 + 0.272i$
Analytic conductor: \(49.9711\)
Root analytic conductor: \(49.9711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 465,\ (1:\ ),\ -0.962 + 0.272i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1105656962 - 0.7948717069i\)
\(L(\frac12)\) \(\approx\) \(-0.1105656962 - 0.7948717069i\)
\(L(1)\) \(\approx\) \(0.5690263872 - 0.3814756198i\)
\(L(1)\) \(\approx\) \(0.5690263872 - 0.3814756198i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 - T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + (0.809 + 0.587i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−24.2361774335469783048090962707, −23.39981179203209581822799731963, −22.74465998334382423174951451519, −21.384473271223235270610209454047, −20.76779013144517205028358298003, −19.39874317643978713718344044723, −19.06407252426618223985478243803, −18.05165673241423277338721963112, −17.37414771215859153639946166578, −16.37113404572856904820061033999, −15.50121708768783970960276467552, −15.00108320330660169678858997508, −13.91744267989228195812780371120, −12.70903200618918082253297828094, −11.79496029143696484281026063135, −10.661350567794018675868559048310, −9.84697769465950242961538484534, −8.90188952621074728680070543907, −8.21651114828991466557470840699, −7.084084729420453724950125073050, −6.149478204474857518445179313875, −5.39484190526757432883229220427, −4.02773110450523546437780715908, −2.38378437484113029795934279960, −1.46503361247506828573131992974, 0.31937537437357994142673361249, 1.07858053997965318212991836080, 2.72300849361912919981769436763, 3.472314880585699164738703082338, 4.66335641101723550658034079187, 6.249994726596739356699520855610, 7.15274964112935769370485610920, 8.19255861939990281822273349447, 8.90925356233759692179434821181, 10.09862403102943769183188527610, 10.762288455765885103924803939576, 11.4413287495837403890259273673, 12.770404790950354654400511005574, 13.32698583372772506817688644887, 14.37703609015291964174410551643, 16.00783807402799731016698587190, 16.241189856005929545331008245147, 17.36477641318224376699073480349, 18.09155752012821914107431515334, 19.04804049308633218188479408488, 19.67968769531975767741917339314, 20.70479410030948522418452470853, 21.10963838854084782807468132671, 22.31709467645931555782208936216, 23.080991633862718796191207522210

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