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Properties

Label 1-61-61.13-r0-0-0
Degree $1$
Conductor $61$
Sign $0.0213 - 0.999i$
Analytic cond. $0.283282$
Root an. cond. $0.283282$
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 + 3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + 9-s + (−0.5 + 0.866i)10-s + 11-s + (−0.5 + 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + 9-s + (−0.5 + 0.866i)10-s + 11-s + (−0.5 + 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0213 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0213 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $0.0213 - 0.999i$
Analytic conductor: \(0.283282\)
Root analytic conductor: \(0.283282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (0:\ ),\ 0.0213 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5916381300 - 0.6044088587i\)
\(L(\frac12)\) \(\approx\) \(0.5916381300 - 0.6044088587i\)
\(L(1)\) \(\approx\) \(0.8079329434 - 0.4961503306i\)
\(L(1)\) \(\approx\) \(0.8079329434 - 0.4961503306i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + T \)
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

−32.82975803053727387215686764395, −31.72143489784037926172869209759, −31.02359903877429984528605660969, −29.54070568254777746955166205059, −27.96212734108874054596461712265, −26.89666436704410662926573769520, −26.15337693423439675395352167939, −25.13490943514034267394366413689, −24.29421379594717280075357689034, −22.77019385383249582309188474141, −21.74108351437040238648104239787, −19.706347272197092488281626733046, −19.137076518533790416768537994577, −18.18021521938183410165258827092, −16.47405153626704325893512285373, −15.19584657111259492259256410243, −14.71187088931571768763326609404, −13.37873363200355781688158435610, −11.4404523107845653887103214409, −9.597018313227289565621270943754, −8.91030542722195412769384214472, −7.364354476297206476628795997476, −6.489227180496968559879100752333, −4.29700410299232839592910764281, −2.47099081751344251375657455374, 1.38715168122094925969059728356, 3.387973917757168832598348089817, 4.33094573657499334774251614185, 7.296891966722697399634078253987, 8.458910098254525333914508465157, 9.444681038240238680660926089304, 10.698284438614819671483454127346, 12.49935234433871057763511382642, 13.12286409118076065033085831922, 14.66149038343365584590602505173, 16.345702719358568942869242671991, 17.33601982044404216945392917780, 19.10962079349383845884442788240, 19.86764054344346141753970632253, 20.40470329232722230340647515635, 21.722623679089190046879205330888, 23.15093036444211428131358296205, 24.70682138019908647821401226929, 25.73407052036152546333017836659, 27.04909731746764861612684530269, 27.51274532280447904608924038998, 29.073094114268846475067897618002, 30.05347748444335830996872545628, 31.018745735794106776256289135781, 32.12337883953066162817471499867

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