↑

Properties

Label 1-605-605.4-r0-0-0
Degree $1$
Conductor $605$
Sign $-0.860 + 0.509i$
Analytic cond. $2.80960$
Root an. cond. $2.80960$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.993 − 0.113i)2-s + (0.809 + 0.587i)3-s + (0.974 + 0.226i)4-s + (−0.736 − 0.676i)6-s + (−0.696 − 0.717i)7-s + (−0.941 − 0.336i)8-s + (0.309 + 0.951i)9-s + (0.654 + 0.755i)12-s + (−0.516 + 0.856i)13-s + (0.610 + 0.791i)14-s + (0.897 + 0.441i)16-s + (−0.774 + 0.633i)17-s + (−0.198 − 0.980i)18-s + (−0.998 − 0.0570i)19-s + (−0.142 − 0.989i)21-s + ⋯
L(s)  = 1  + (−0.993 − 0.113i)2-s + (0.809 + 0.587i)3-s + (0.974 + 0.226i)4-s + (−0.736 − 0.676i)6-s + (−0.696 − 0.717i)7-s + (−0.941 − 0.336i)8-s + (0.309 + 0.951i)9-s + (0.654 + 0.755i)12-s + (−0.516 + 0.856i)13-s + (0.610 + 0.791i)14-s + (0.897 + 0.441i)16-s + (−0.774 + 0.633i)17-s + (−0.198 − 0.980i)18-s + (−0.998 − 0.0570i)19-s + (−0.142 − 0.989i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.860 + 0.509i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ -0.860 + 0.509i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1255668703 + 0.4582938423i\)
\(L(\frac12)\) \(\approx\) \(0.1255668703 + 0.4582938423i\)
\(L(1)\) \(\approx\) \(0.6365787850 + 0.1866300637i\)
\(L(1)\) \(\approx\) \(0.6365787850 + 0.1866300637i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.993 - 0.113i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (-0.696 - 0.717i)T \)
13 \( 1 + (-0.516 + 0.856i)T \)
17 \( 1 + (-0.774 + 0.633i)T \)
19 \( 1 + (-0.998 - 0.0570i)T \)
23 \( 1 + (0.142 - 0.989i)T \)
29 \( 1 + (-0.362 + 0.931i)T \)
31 \( 1 + (-0.921 - 0.389i)T \)
37 \( 1 + (-0.0855 + 0.996i)T \)
41 \( 1 + (-0.870 + 0.491i)T \)
43 \( 1 + (0.959 - 0.281i)T \)
47 \( 1 + (-0.198 + 0.980i)T \)
53 \( 1 + (-0.897 + 0.441i)T \)
59 \( 1 + (-0.870 - 0.491i)T \)
61 \( 1 + (0.993 - 0.113i)T \)
67 \( 1 + (-0.415 + 0.909i)T \)
71 \( 1 + (-0.254 - 0.967i)T \)
73 \( 1 + (0.466 + 0.884i)T \)
79 \( 1 + (-0.0285 - 0.999i)T \)
83 \( 1 + (0.985 + 0.170i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (0.564 + 0.825i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.84542867196973355989469761490, −21.68297800610482375648305764119, −20.80865852887549634711159049567, −19.86020997892713673057377435869, −19.44281787993189464979386925426, −18.64499479723561780955487246381, −17.90418376542655981801441560365, −17.15969552442789635445548601047, −15.89978088336763816431117675786, −15.328065942406597332909035149, −14.60424258439138835797964439348, −13.308734995422318821089670285417, −12.56412914790130419242114041804, −11.71604824101467344175983515521, −10.551608534913550605926948432829, −9.48565123056469498104343162511, −9.031194644163388798833571692350, −8.07673319004193824209767052507, −7.25588195893110726135333497656, −6.43599793380014000597299575292, −5.455336302549059439001153987113, −3.59131440182043203351921522371, −2.60153088535178894549936390779, −1.90902537006727552151062420922, −0.27088868785375650795455211854, 1.67912195098482597819402860405, 2.63780444216164826319944091973, 3.69400852643402451023277345762, 4.59609926956972062476302909389, 6.35468091792143714141767880750, 7.06604987242758095392363427102, 8.0791622399673156957489964840, 8.96845226685385608622248207089, 9.562213338252240569999836572243, 10.53496848114870103089718102994, 10.996445277695736184408667048729, 12.45687287715302784138746410134, 13.24264288075915885157421447915, 14.44704249274091390264124271952, 15.118208409024945309862575591322, 16.15613310777446648352627842215, 16.68449549506872252664203159252, 17.442576952484749555927680915588, 18.88547393308775015982172977572, 19.163405758718051510252873359610, 20.17641439591797043745086174701, 20.52106091984471953825584987410, 21.68510397390399454189856570293, 22.19433566032215237109128874879, 23.706829640134916736734715006072

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