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Properties

Label 1-65-65.62-r1-0-0
Degree $1$
Conductor $65$
Sign $0.0171 + 0.999i$
Analytic cond. $6.98522$
Root an. cond. $6.98522$
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  + (0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (0.5 + 0.866i)6-s + (−0.866 + 0.5i)7-s + i·8-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s + i·12-s − 14-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + i·18-s + (−0.5 − 0.866i)19-s − 21-s + (0.866 − 0.5i)22-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (0.5 + 0.866i)6-s + (−0.866 + 0.5i)7-s + i·8-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s + i·12-s − 14-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + i·18-s + (−0.5 − 0.866i)19-s − 21-s + (0.866 − 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0171 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0171 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.0171 + 0.999i$
Analytic conductor: \(6.98522\)
Root analytic conductor: \(6.98522\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 65,\ (1:\ ),\ 0.0171 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.249724207 + 2.288646014i\)
\(L(\frac12)\) \(\approx\) \(2.249724207 + 2.288646014i\)
\(L(1)\) \(\approx\) \(1.825392141 + 1.096057620i\)
\(L(1)\) \(\approx\) \(1.825392141 + 1.096057620i\)

Euler product

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

−31.56733966400274543865730556938, −30.53483143382970417224109288521, −29.75342494666273244244984699089, −28.776771984987788404715963620, −27.292450410688633343482267483005, −25.69832434165344004580347315842, −25.04606640896157265546834127392, −23.60892045495582333100285387184, −22.87200424783546251838575198196, −21.4161770365663677715069548468, −20.27446349856329078906157138611, −19.5596382812560942961625569573, −18.51767765710768460115986726775, −16.5852300770226784170445308827, −15.01973658292010972779024108469, −14.25110095969256723499420647103, −12.923469126131354177380626259873, −12.34012161237399847261535312751, −10.43241279832577950755577022290, −9.3209872855836719083292811682, −7.3731994108314045717850047015, −6.24980626020211210777476806923, −4.18429687264048783691262882202, −3.04153665518664138424644836759, −1.41841578871220177691053335801, 2.71724135842283025702547344112, 3.726164208189032379512948789490, 5.33351914546999686482740968010, 6.815594797824762417950401332163, 8.33181454803260980559803328445, 9.49369928701863899017869754872, 11.31465875993765833625400112332, 12.85237567380308097534444383494, 13.8180846867922974659853949615, 14.951884115250951352014534871967, 15.90868370467059622707469093069, 16.85012123218156404456542388824, 18.90781158292256360289073745742, 19.97888391005019466087818863134, 21.345698215099120375950068626726, 21.9612015326576118827870125637, 23.21933368782199879437979934161, 24.67677926436738267427346883705, 25.41130804783354452532612703444, 26.33354260406715261382965991323, 27.5133683769114671148234757915, 29.21993638740256696401530818286, 30.30854400282240089804106938105, 31.45197266258040901778853772740, 32.2266749119683817613012466739

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