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Properties

Label 1-65-65.28-r0-0-0
Degree $1$
Conductor $65$
Sign $-0.892 - 0.450i$
Analytic cond. $0.301858$
Root an. cond. $0.301858$
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  + (0.5 − 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.5 − 0.866i)4-s + (−0.866 + 0.5i)6-s + (−0.5 − 0.866i)7-s − 8-s + (0.5 + 0.866i)9-s + (−0.866 − 0.5i)11-s + i·12-s − 14-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + 18-s + (0.866 − 0.5i)19-s + i·21-s + (−0.866 + 0.5i)22-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.5 − 0.866i)4-s + (−0.866 + 0.5i)6-s + (−0.5 − 0.866i)7-s − 8-s + (0.5 + 0.866i)9-s + (−0.866 − 0.5i)11-s + i·12-s − 14-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + 18-s + (0.866 − 0.5i)19-s + i·21-s + (−0.866 + 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.892 - 0.450i$
Analytic conductor: \(0.301858\)
Root analytic conductor: \(0.301858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 65,\ (0:\ ),\ -0.892 - 0.450i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1725111420 - 0.7239896890i\)
\(L(\frac12)\) \(\approx\) \(0.1725111420 - 0.7239896890i\)
\(L(1)\) \(\approx\) \(0.5862291025 - 0.6490779799i\)
\(L(1)\) \(\approx\) \(0.5862291025 - 0.6490779799i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 - iT \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + T \)
53 \( 1 + iT \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−32.78292543221838080675748853695, −31.817609007458293948573397459784, −30.779088468898706337867895306784, −29.21709930053563622937655871071, −28.22422212147210711438830832167, −27.03511621554435412931251442701, −25.93430598145004397883269969786, −24.8620284569405074892484688638, −23.52526746586371647635035981290, −22.790448954747694767150142030898, −21.75750312505337268307336231722, −20.86763340121065281045189961871, −18.63918258212224936753745623636, −17.686540776724996477859818853571, −16.38046981460857651173199775745, −15.67982426680774492362426657935, −14.58340309196046688499030500483, −12.77732606086071245553232255290, −12.08006788222770602617352704464, −10.27707251802143623707070451031, −8.892977632768086371668516279043, −7.19898315066022079145563888485, −5.82564357269611187036892591201, −4.99502032582193257113854746834, −3.28950424031960333749489515682, 0.916465872791627052296361685578, 3.01085226967803786276873082992, 4.77348025942900318642882436964, 5.99412281866392273152516566426, 7.53291390951785008194723940770, 9.75242760834868304865862804814, 10.80707099954993125327097401024, 11.83305911679966408647743277668, 13.136801358915376159384082259561, 13.779984654256255858312347764830, 15.71398246966678775132819474053, 17.047613155116009878504223347043, 18.38516536880591666976598446489, 19.24789115516146738438500238414, 20.551807503976313709418292969159, 21.74290154948683472441933470451, 22.94064606938467912568856925173, 23.47690348177033166958844720488, 24.63201096854866407282312384068, 26.48491309446251710178283655059, 27.66734672383564929455851819845, 28.85394053738842424945659770174, 29.44367486655608077494406050323, 30.34352935743382190064609879352, 31.55236168746130405770935687255

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