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Properties

Label 1-65-65.37-r0-0-0
Degree $1$
Conductor $65$
Sign $0.558 - 0.829i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.558 - 0.829i$
Analytic conductor: \(0.301858\)
Root analytic conductor: \(0.301858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 65,\ (0:\ ),\ 0.558 - 0.829i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.178015653 - 0.6272869773i\)
\(L(\frac12)\) \(\approx\) \(1.178015653 - 0.6272869773i\)
\(L(1)\) \(\approx\) \(1.329146014 - 0.5176189614i\)
\(L(1)\) \(\approx\) \(1.329146014 - 0.5176189614i\)

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

−32.09067196537610369377069882642, −31.59302624168860709357857854620, −30.44877137361678713789581932957, −29.4633468016934265124873033449, −27.66998363120581024536878007517, −26.41547675774510436895248212241, −25.48525734594779389682652571328, −24.71189315357774683614900184900, −23.80789536221462709511839136860, −22.3256800343610693133995461709, −21.45353518634732844345457653861, −19.89342809168939886249380489039, −18.72625533436846968171592194190, −17.58156347587138152507022752543, −16.03049339587402112884782395580, −15.07106622569083380019152575126, −13.9828509720996818674155739844, −12.96176612488092835741955269921, −11.85451602208834916555844633593, −9.26430873071912295131283421734, −8.54763302124368413989164609689, −7.02783866058890490811283571108, −5.99231753293853993282316175600, −4.04274191597153076305037619849, −2.6118409976260703212680330680, 1.990644712139968536190347235303, 3.65554214121141898285561090791, 4.46440976885799827409227494486, 6.58637150157554922505458887056, 8.56802522234116986467888290061, 9.84908569999033472436854271873, 10.66750390250397642447134832182, 12.37283955541092004913660395934, 13.58104135887609002833703597485, 14.446699067819491704139956123370, 15.63422483904511053428816383629, 17.26082001613647738926941679829, 19.04740413810826280078539827771, 19.86453090215839358994880287908, 20.61540909371875361863487306003, 21.86180179593241836116632702065, 22.76791678887855322537208005778, 24.10174400397860353746541292087, 25.481635763844114595756164132287, 26.71331559496391397741344655463, 27.60193636309719416786123685246, 28.846946711011636366690926741999, 30.15965966143359195874511238179, 30.74313918167800422942627284280, 32.10915188254526851577000469442

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