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Properties

Label 1-1205-1205.1132-r0-0-0
Degree $1$
Conductor $1205$
Sign $0.763 - 0.646i$
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 + (−0.760 + 0.649i)7-s + (0.707 + 0.707i)8-s + (−0.951 − 0.309i)9-s + (−0.923 + 0.382i)11-s + (−0.987 − 0.156i)12-s + (0.233 + 0.972i)13-s + (0.0784 − 0.996i)14-s − 16-s + (−0.996 − 0.0784i)17-s + (0.891 − 0.453i)18-s + (0.382 + 0.923i)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 + (−0.760 + 0.649i)7-s + (0.707 + 0.707i)8-s + (−0.951 − 0.309i)9-s + (−0.923 + 0.382i)11-s + (−0.987 − 0.156i)12-s + (0.233 + 0.972i)13-s + (0.0784 − 0.996i)14-s − 16-s + (−0.996 − 0.0784i)17-s + (0.891 − 0.453i)18-s + (0.382 + 0.923i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.763 - 0.646i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (1132, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ 0.763 - 0.646i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5694785408 - 0.2087749880i\)
\(L(\frac12)\) \(\approx\) \(0.5694785408 - 0.2087749880i\)
\(L(1)\) \(\approx\) \(0.6109601064 + 0.006992135430i\)
\(L(1)\) \(\approx\) \(0.6109601064 + 0.006992135430i\)

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 \( 1 + (-0.760 + 0.649i)T \)
11 \( 1 + (-0.923 + 0.382i)T \)
13 \( 1 + (0.233 + 0.972i)T \)
17 \( 1 + (-0.996 - 0.0784i)T \)
19 \( 1 + (0.382 + 0.923i)T \)
23 \( 1 + (-0.852 - 0.522i)T \)
29 \( 1 + (-0.453 - 0.891i)T \)
31 \( 1 + (0.0784 - 0.996i)T \)
37 \( 1 + (-0.233 + 0.972i)T \)
41 \( 1 + (0.453 - 0.891i)T \)
43 \( 1 + (0.649 - 0.760i)T \)
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 \( 1 + (0.649 - 0.760i)T \)
73 \( 1 + (0.233 - 0.972i)T \)
79 \( 1 + (0.156 + 0.987i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (0.923 + 0.382i)T \)
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

−21.21238548552764852806499091751, −20.20151507063738846789739306319, −19.97732166674976046346524522280, −19.25809568982311381514887236809, −17.95863722761113481905570383826, −17.67497365644450569976652872569, −16.55056468364844699414318100605, −15.91722259192547821084044616726, −15.55599958742337804924269100837, −14.15213383295102396369718477624, −13.28096178004614690813716823588, −12.79249129082673297320190833982, −11.50437307764033282447639656907, −10.7140827708376087799653436772, −10.41756579165718257175064099478, −9.483101893716022821339245133799, −8.84083953040419764400356415792, −7.953366589818572498368117210292, −7.12739860853663817376225893428, −5.86472004428756275988127669620, −4.80382612964525069577151687378, −3.827949063182631912676513427488, −3.13016224825853650592938555870, −2.41834400581566499926195942929, −0.74404186990457430190864679803, 0.433070081323771063974833355157, 2.028443770296680847195669535379, 2.344537892106372990172610278723, 3.954599620936048534571816761786, 5.30522230574912704633785099836, 6.11163229196276095908291022699, 6.65111872253688014734625863903, 7.5716310516255044605267612132, 8.24507769328795930499628832874, 9.08894143674522711863158090557, 9.74149108336451483008638433830, 10.787083964600137038500771139392, 11.780835759946506095301446990, 12.52761201766479647983369019249, 13.54808439668299809432451036156, 13.99082798703517683547363178230, 15.145354894720836875364573535030, 15.68092626543371247294704420112, 16.557096394444537975774874456, 17.31755060970431933232095342060, 18.22331664025069280363364424992, 18.74052977120819485369056814206, 19.08799803654294661714814657097, 20.18302510423551568935022613037, 20.690576068030758589261206319503

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