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Properties

Label 1-1205-1205.1112-r0-0-0
Degree $1$
Conductor $1205$
Sign $0.643 + 0.765i$
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  + (0.707 − 0.707i)2-s + (−0.156 + 0.987i)3-s − i·4-s + (0.587 + 0.809i)6-s + (0.649 + 0.760i)7-s + (−0.707 − 0.707i)8-s + (−0.951 − 0.309i)9-s + (0.382 + 0.923i)11-s + (0.987 + 0.156i)12-s + (−0.972 + 0.233i)13-s + (0.996 + 0.0784i)14-s − 16-s + (0.0784 − 0.996i)17-s + (−0.891 + 0.453i)18-s + (0.923 − 0.382i)19-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.156 + 0.987i)3-s − i·4-s + (0.587 + 0.809i)6-s + (0.649 + 0.760i)7-s + (−0.707 − 0.707i)8-s + (−0.951 − 0.309i)9-s + (0.382 + 0.923i)11-s + (0.987 + 0.156i)12-s + (−0.972 + 0.233i)13-s + (0.996 + 0.0784i)14-s − 16-s + (0.0784 − 0.996i)17-s + (−0.891 + 0.453i)18-s + (0.923 − 0.382i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.643 + 0.765i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (1112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ 0.643 + 0.765i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.749641740 + 0.8152556644i\)
\(L(\frac12)\) \(\approx\) \(1.749641740 + 0.8152556644i\)
\(L(1)\) \(\approx\) \(1.416271296 + 0.08310226425i\)
\(L(1)\) \(\approx\) \(1.416271296 + 0.08310226425i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.156 + 0.987i)T \)
7 \( 1 + (0.649 + 0.760i)T \)
11 \( 1 + (0.382 + 0.923i)T \)
13 \( 1 + (-0.972 + 0.233i)T \)
17 \( 1 + (0.0784 - 0.996i)T \)
19 \( 1 + (0.923 - 0.382i)T \)
23 \( 1 + (-0.522 + 0.852i)T \)
29 \( 1 + (0.453 + 0.891i)T \)
31 \( 1 + (0.996 + 0.0784i)T \)
37 \( 1 + (0.972 + 0.233i)T \)
41 \( 1 + (-0.453 + 0.891i)T \)
43 \( 1 + (0.760 + 0.649i)T \)
47 \( 1 + (-0.891 + 0.453i)T \)
53 \( 1 + (-0.453 - 0.891i)T \)
59 \( 1 + (0.156 + 0.987i)T \)
61 \( 1 + (-0.156 + 0.987i)T \)
67 \( 1 + (-0.891 - 0.453i)T \)
71 \( 1 + (0.760 + 0.649i)T \)
73 \( 1 + (-0.972 - 0.233i)T \)
79 \( 1 + (-0.156 - 0.987i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (-0.382 + 0.923i)T \)
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

−21.1842849845512437371659936233, −20.31393616275419934317971370198, −19.57412092526018509713567859834, −18.65568270804396144320182323274, −17.73690034487994930860755856874, −17.10606091230531253885197476547, −16.707885212626029881594127714434, −15.61917174136097977978627165645, −14.491801694898093839169244341473, −14.14923544052315057464071945507, −13.47291314699585059435917532501, −12.57705176239822396571081606443, −11.9073523557233974939573040875, −11.2199683159439793623386251990, −10.14605277323130746681961868801, −8.701443143768783073463629301778, −7.9752357033968088100898994516, −7.52795413099098759875708775369, −6.509386506994917910293017180122, −5.919429228644038614936204991157, −4.97659850824758285421226352025, −4.03616453656063598053218704588, −3.01773669631030471550065532518, −1.96986412608277415981574470094, −0.645967843627719320978557105566, 1.300311250789807585933581324037, 2.51211777123007775784021171451, 3.11238418125560413706042120007, 4.44656910283857836225333442341, 4.82087817042245816445486462614, 5.52050360576597630647026946492, 6.58094423675722828641932817945, 7.74022806351603064968275319154, 9.11124004437389569998100524826, 9.58257316656566652636025813693, 10.21270136439347078955649956972, 11.48000195085100766312215528345, 11.67782904543856405935904175002, 12.40874052655609868546896708490, 13.64934778116449112281490482710, 14.48288323031252653740931370690, 14.89578055301854732213370684351, 15.68369423105909651325136966402, 16.40809791337395715663503175965, 17.79958275115485666970080936274, 17.96046859895624763254519622668, 19.357921257437527775954919751685, 19.96925544439030719995383384692, 20.65894464449673964257868942998, 21.35045541974995099119421198150

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