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Properties

Label 1-65-65.8-r0-0-0
Degree $1$
Conductor $65$
Sign $0.749 + 0.661i$
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  + 2-s + i·3-s + 4-s + i·6-s − 7-s + 8-s − 9-s − i·11-s + i·12-s − 14-s + 16-s + i·17-s − 18-s − i·19-s − i·21-s − i·22-s + ⋯
L(s)  = 1  + 2-s + i·3-s + 4-s + i·6-s − 7-s + 8-s − 9-s − i·11-s + i·12-s − 14-s + 16-s + i·17-s − 18-s − i·19-s − i·21-s − i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.749 + 0.661i$
Analytic conductor: \(0.301858\)
Root analytic conductor: \(0.301858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 65,\ (0:\ ),\ 0.749 + 0.661i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.370223614 + 0.5182767396i\)
\(L(\frac12)\) \(\approx\) \(1.370223614 + 0.5182767396i\)
\(L(1)\) \(\approx\) \(1.509020967 + 0.3934177522i\)
\(L(1)\) \(\approx\) \(1.509020967 + 0.3934177522i\)

Euler product

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

−31.67672906426039715577274133851, −31.18225832852193301584617457271, −29.77160546293504586129371889553, −29.3055513243783886595917395257, −28.100492331750352999294686111205, −25.92799470888533723557828048136, −25.260895045787242570960807399733, −24.20244069896380307661487297243, −22.92684144059871440778320203568, −22.5710746238596995168529542523, −20.7573108308320364486410852110, −19.794610609711137157839077243726, −18.70245836868697524493959515176, −17.15407976702283749818794367538, −15.87555225541885527284388745111, −14.52411608373154521630155619692, −13.34959280200627499809914976107, −12.539180530275797883243500302988, −11.51603978912734431588818044882, −9.7362230751714873125686076329, −7.62449161914241303510973803601, −6.681703304601103134456438392618, −5.43063343920782791257483816656, −3.50131320946130805990772191375, −2.021772809675842337552614596094, 2.90412928662050702441472068580, 3.9481074453100913983968138706, 5.443838858444506126727971709034, 6.59275479542356088164165191934, 8.657566982297700079000366442275, 10.24367035547808864852972342042, 11.23379464835647882019909554642, 12.71520260351638053829560741744, 13.91165143025147292765528815983, 15.13876593942303081398713292265, 16.0912078599684542591068481651, 16.937287913756485956335891842757, 19.223035996993874458235487904242, 20.188764571321683522745992388904, 21.48043337481769483807996723489, 22.11685481116609652099043754270, 23.13844817907205670553600961034, 24.384732968951227329665134656671, 25.78732605882320498162277696973, 26.52083949374780001601147821427, 28.201631990661722202553890557307, 29.025443496265801308883278279476, 30.25796209793602351175812045235, 31.5758704623335932514376905170, 32.41853970777710965853987771892

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