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Properties

Label 1-61-61.35-r1-0-0
Degree $1$
Conductor $61$
Sign $-0.378 + 0.925i$
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  + (0.406 + 0.913i)2-s + (0.809 + 0.587i)3-s + (−0.669 + 0.743i)4-s + (0.978 + 0.207i)5-s + (−0.207 + 0.978i)6-s + (0.994 − 0.104i)7-s + (−0.951 − 0.309i)8-s + (0.309 + 0.951i)9-s + (0.207 + 0.978i)10-s − i·11-s + (−0.978 + 0.207i)12-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (0.669 + 0.743i)15-s + (−0.104 − 0.994i)16-s + (−0.743 − 0.669i)17-s + ⋯
L(s)  = 1  + (0.406 + 0.913i)2-s + (0.809 + 0.587i)3-s + (−0.669 + 0.743i)4-s + (0.978 + 0.207i)5-s + (−0.207 + 0.978i)6-s + (0.994 − 0.104i)7-s + (−0.951 − 0.309i)8-s + (0.309 + 0.951i)9-s + (0.207 + 0.978i)10-s − i·11-s + (−0.978 + 0.207i)12-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (0.669 + 0.743i)15-s + (−0.104 − 0.994i)16-s + (−0.743 − 0.669i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $-0.378 + 0.925i$
Analytic conductor: \(6.55536\)
Root analytic conductor: \(6.55536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (1:\ ),\ -0.378 + 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.536939708 + 2.289082429i\)
\(L(\frac12)\) \(\approx\) \(1.536939708 + 2.289082429i\)
\(L(1)\) \(\approx\) \(1.417020081 + 1.199564842i\)
\(L(1)\) \(\approx\) \(1.417020081 + 1.199564842i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 \)
good2 \( 1 + (0.406 + 0.913i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (0.978 + 0.207i)T \)
7 \( 1 + (0.994 - 0.104i)T \)
11 \( 1 - iT \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.743 - 0.669i)T \)
19 \( 1 + (0.104 - 0.994i)T \)
23 \( 1 + (-0.951 + 0.309i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (0.406 - 0.913i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + (0.743 - 0.669i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.951 + 0.309i)T \)
59 \( 1 + (-0.406 - 0.913i)T \)
67 \( 1 + (-0.207 + 0.978i)T \)
71 \( 1 + (0.207 + 0.978i)T \)
73 \( 1 + (-0.978 + 0.207i)T \)
79 \( 1 + (-0.743 + 0.669i)T \)
83 \( 1 + (0.913 - 0.406i)T \)
89 \( 1 + (-0.587 + 0.809i)T \)
97 \( 1 + (-0.913 - 0.406i)T \)
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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

−31.69141843121817964242923338692, −30.592759802919120161298716029404, −29.99674631983953027765839845680, −28.85438055713974941503960499551, −27.73018508148762692942626802624, −26.35688474815924714822839606954, −24.92873294296256029430117063616, −24.233942161894382784308769745330, −22.7485911871717661903076194550, −21.37404596646871393498440831749, −20.575784312245981307230066269619, −19.71606517319558049341063516365, −18.09782461056520860896348833418, −17.6838688810108916931895351783, −14.95371778776520862003955578270, −14.27529829157013021422953845450, −13.03938090904368326685687879802, −12.22195565784487685774170191620, −10.431342143966585251967190079068, −9.31955498840320541462857502974, −7.956039858166477769365811552295, −5.92951717148108015290195773752, −4.385186764255375996056803801086, −2.43496440388094088528916070684, −1.53170316391269927953248781195, 2.47719855403532062615492293512, 4.27492447017180174705587498958, 5.481363363155096883301024852182, 7.158904625889116964446562653067, 8.57819174200859028203137691161, 9.525112179225768471489774871888, 11.28529021865090868263275744054, 13.49296807424794475098140541642, 14.01047882613044723374973682214, 15.01495353289169671495376058488, 16.32441884843216366556394329325, 17.44155573243131668820252416883, 18.67052741096260339483176213176, 20.502919057467707512130935722908, 21.63834131131065890944092880692, 22.06590931123849852629482364916, 24.14881957983253976829149174851, 24.63961818045445922648604268204, 26.048977358927637206634206713479, 26.57299158227935192577800181234, 27.74202401944758579632859865139, 29.61089784698422932857867514249, 30.75489907768398848029506030819, 31.70921923761483109943905190577, 32.69586388435692861009007799760

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