Properties

Label 1-61-61.11-r1-0-0
Degree $1$
Conductor $61$
Sign $0.905 - 0.424i$
Analytic cond. $6.55536$
Root an. cond. $6.55536$
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  + i·2-s − 3-s − 4-s − 5-s i·6-s + i·7-s i·8-s + 9-s i·10-s i·11-s + 12-s + 13-s − 14-s + 15-s + 16-s i·17-s + ⋯
L(s)  = 1  + i·2-s − 3-s − 4-s − 5-s i·6-s + i·7-s i·8-s + 9-s i·10-s i·11-s + 12-s + 13-s − 14-s + 15-s + 16-s i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $0.905 - 0.424i$
Analytic conductor: \(6.55536\)
Root analytic conductor: \(6.55536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (1:\ ),\ 0.905 - 0.424i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4892641900 - 0.1089028119i\)
\(L(\frac12)\) \(\approx\) \(0.4892641900 - 0.1089028119i\)
\(L(1)\) \(\approx\) \(0.5348761848 + 0.1936518880i\)
\(L(1)\) \(\approx\) \(0.5348761848 + 0.1936518880i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + iT \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - iT \)
19 \( 1 + iT \)
23 \( 1 - iT \)
29 \( 1 + T \)
31 \( 1 - iT \)
37 \( 1 - iT \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 + T \)
67 \( 1 + iT \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 - iT \)
83 \( 1 + T \)
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

−32.344073113395055618320788873581, −30.652778309095281356457432754179, −30.314024467842600048491901892557, −28.90144749255012295283502482328, −27.99174234186161321801617345602, −27.27657126700586062091939098852, −26.07911622612628338756373134907, −23.654728512623868509951398559180, −23.3558755858754926177142675545, −22.320557411311411884309558106814, −20.901084430959454503525235156096, −19.89319417889539657248986443703, −18.7127923856408314336749180057, −17.55466936162382183101351609354, −16.4591744170512024707848276403, −14.87180284544023961642099983005, −13.06891246141014263097838946690, −12.21978490992863552288543159151, −10.92685206990001004567321884594, −10.33687435358757835635991922637, −8.35857067480103636685037099785, −6.75744902898750149480230858809, −4.68446139332045165546829606238, −3.82361314037827472262379881245, −1.25744516575100415810130807668, 0.36365065701657626555964807386, 3.87494521687023205064231515912, 5.37365431713090252341023294940, 6.392173076998325769877167790385, 7.87145800300535819715638960152, 9.10849848754717341921524354062, 11.04107349743961666807273523793, 12.12632568681488460361038074326, 13.502354980208955116720302855792, 15.347467268455097293251658771476, 15.88022926942884704149935356090, 16.97682935860903540549019281984, 18.43509501717034314923118485246, 18.98064319788907955241976117164, 21.30676902723992398529211477548, 22.40718844930462295984200111417, 23.34825002217911036063958768431, 24.15168161841516872392111429825, 25.2776189399252214116827495461, 26.81826940092410395103687192334, 27.65273610374729114818307187257, 28.422813818855682239491253693093, 30.07636183277276763158644102184, 31.457576830032217557270788722308, 32.18594394557742149206274874373

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