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Properties

Label 1-61-61.34-r0-0-0
Degree $1$
Conductor $61$
Sign $-0.754 - 0.656i$
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  + (0.309 − 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 − 0.587i)4-s + (−0.809 − 0.587i)5-s + (−0.809 − 0.587i)6-s + (0.309 + 0.951i)7-s + (−0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.809 + 0.587i)10-s + 11-s + (−0.809 + 0.587i)12-s + 13-s + 14-s + (−0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + (−0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 − 0.587i)4-s + (−0.809 − 0.587i)5-s + (−0.809 − 0.587i)6-s + (0.309 + 0.951i)7-s + (−0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.809 + 0.587i)10-s + 11-s + (−0.809 + 0.587i)12-s + 13-s + 14-s + (−0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + (−0.809 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $-0.754 - 0.656i$
Analytic conductor: \(0.283282\)
Root analytic conductor: \(0.283282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (0:\ ),\ -0.754 - 0.656i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3366697500 - 0.9000447900i\)
\(L(\frac12)\) \(\approx\) \(0.3366697500 - 0.9000447900i\)
\(L(1)\) \(\approx\) \(0.7136636963 - 0.8068663094i\)
\(L(1)\) \(\approx\) \(0.7136636963 - 0.8068663094i\)

Euler product

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

−33.20004189050527484517910868407, −32.067969312896579609109994143370, −30.94291977122332857480719686076, −30.211524131378885594272217616840, −27.932931710482086211769450543277, −27.05982174884974215765435093424, −26.41200997314718902680158610125, −25.34455561351558303910704438050, −23.84476807853839908769292671249, −22.875453570397195967175559838900, −22.023826232493723733048177982523, −20.62113443893446303171615606751, −19.39656583561067003352591352813, −17.71549765858510759112672510157, −16.512573170863572036558428245224, −15.59842726808232835153919655348, −14.529258610673864988710000227736, −13.722383003180422810245067439546, −11.67141514953135441879634467162, −10.34267328434728078710014142935, −8.73080463602224800186174647764, −7.64052940147192160685204498205, −6.16202969896536879588572819781, −4.19143424574844943564633835213, −3.71354775722383375085340538131, 1.30609547233326499173689447426, 2.95575865137151674089598088487, 4.608863603497329265041014590609, 6.34442599676768891234968307662, 8.39676766752661059040362586458, 9.06486329043872843979979518291, 11.42351624623166560449332570575, 11.96659412985839089272678285298, 13.11399775303122040727644730567, 14.29281758725641489736133534604, 15.64627253656881302846196172778, 17.69928150415130183064343303639, 18.64312804567581645278216637812, 19.72740568090451960527118326794, 20.41292821141916634201521282743, 21.90753394125294249491637051972, 23.143409326237150101487387677, 24.15377562946566536840402098960, 25.04689679238848056313040934388, 26.84036188592062438761128201144, 28.122879939212476262508380598333, 28.68884814568326082550867574245, 30.26127818293140486106231611704, 30.82693552603964077243794419651, 31.70237397031972089456200716390

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