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Properties

Label 1-1205-1205.1004-r0-0-0
Degree $1$
Conductor $1205$
Sign $0.979 - 0.200i$
Analytic cond. $5.59599$
Root an. cond. $5.59599$
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  + 2-s + (−0.809 + 0.587i)3-s + 4-s + (−0.809 + 0.587i)6-s + (−0.951 + 0.309i)7-s + 8-s + (0.309 − 0.951i)9-s − i·11-s + (−0.809 + 0.587i)12-s + (0.587 + 0.809i)13-s + (−0.951 + 0.309i)14-s + 16-s + (0.951 − 0.309i)17-s + (0.309 − 0.951i)18-s − i·19-s + ⋯
L(s)  = 1  + 2-s + (−0.809 + 0.587i)3-s + 4-s + (−0.809 + 0.587i)6-s + (−0.951 + 0.309i)7-s + 8-s + (0.309 − 0.951i)9-s − i·11-s + (−0.809 + 0.587i)12-s + (0.587 + 0.809i)13-s + (−0.951 + 0.309i)14-s + 16-s + (0.951 − 0.309i)17-s + (0.309 − 0.951i)18-s − i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.979 - 0.200i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (1004, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ 0.979 - 0.200i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.090182976 - 0.2119174300i\)
\(L(\frac12)\) \(\approx\) \(2.090182976 - 0.2119174300i\)
\(L(1)\) \(\approx\) \(1.487992600 + 0.08051577148i\)
\(L(1)\) \(\approx\) \(1.487992600 + 0.08051577148i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 - iT \)
13 \( 1 + (0.587 + 0.809i)T \)
17 \( 1 + (0.951 - 0.309i)T \)
19 \( 1 - iT \)
23 \( 1 + (-0.587 - 0.809i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (-0.951 + 0.309i)T \)
37 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + (-0.951 - 0.309i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.309 + 0.951i)T \)
71 \( 1 + (0.951 + 0.309i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + iT \)
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

−21.37154478541638323448533260206, −20.425564402865304309079801039227, −19.83349327907845380135958544797, −18.93397278138697669245708230270, −18.17734788776114828868773979231, −17.083843125450682432771149024815, −16.603679448655388337635809835605, −15.77290891056596431867048053255, −15.04390889871133878336231100161, −14.022447834111375047055011399783, −13.14305417022498012418837269875, −12.76546584909565177163561651001, −12.03741245790676350231530565234, −11.27811643773651702607353486872, −10.21077309451887748331458636413, −9.869627493837775672448535200387, −7.89111523668833985020099228070, −7.54347289629942726522487703378, −6.41400225137908461768269154930, −5.964535273109854050230406645929, −5.16330077790315117845461142852, −4.05126318938827967155563107192, −3.298052981687726814021435483117, −2.07159487949046522559711777968, −1.13252619179314219328734103111, 0.75602953736917362287928065137, 2.28268371113964457202879543860, 3.52835148230977989521977343043, 3.79075859360806950976676021433, 5.153188880998001778613138908550, 5.61356593644499418345388482771, 6.54953783805330146418883473859, 6.99938921316835603349825288189, 8.558145694850385338987992797679, 9.44226514495141769885713453199, 10.41599517792467903857018670537, 11.09789598669013095973110509558, 11.81870227897230911401674888719, 12.51934690215341571411686140254, 13.32134800752208746920195727349, 14.16906895102452016481243128417, 14.99556041569585159224513586870, 15.94419731068313846722823632139, 16.372736509408150193307142858148, 16.74850845635058151505251570938, 18.16029370109322263559533777322, 18.879677270993518813754716536012, 19.79141486957127378635562717604, 20.67124488278665470988686249425, 21.463952255626978284371326878092

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