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Properties

Label 1-1205-1205.1007-r0-0-0
Degree $1$
Conductor $1205$
Sign $-0.959 - 0.281i$
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 + (0.453 − 0.891i)3-s + i·4-s + (−0.951 + 0.309i)6-s + (0.522 + 0.852i)7-s + (0.707 − 0.707i)8-s + (−0.587 − 0.809i)9-s + (0.382 − 0.923i)11-s + (0.891 + 0.453i)12-s + (−0.649 − 0.760i)13-s + (0.233 − 0.972i)14-s − 16-s + (−0.972 − 0.233i)17-s + (−0.156 + 0.987i)18-s + (0.923 + 0.382i)19-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (0.453 − 0.891i)3-s + i·4-s + (−0.951 + 0.309i)6-s + (0.522 + 0.852i)7-s + (0.707 − 0.707i)8-s + (−0.587 − 0.809i)9-s + (0.382 − 0.923i)11-s + (0.891 + 0.453i)12-s + (−0.649 − 0.760i)13-s + (0.233 − 0.972i)14-s − 16-s + (−0.972 − 0.233i)17-s + (−0.156 + 0.987i)18-s + (0.923 + 0.382i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $-0.959 - 0.281i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ -0.959 - 0.281i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1485513381 - 1.032801304i\)
\(L(\frac12)\) \(\approx\) \(0.1485513381 - 1.032801304i\)
\(L(1)\) \(\approx\) \(0.6594688819 - 0.5550508071i\)
\(L(1)\) \(\approx\) \(0.6594688819 - 0.5550508071i\)

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 \( 1 + (0.453 - 0.891i)T \)
7 \( 1 + (0.522 + 0.852i)T \)
11 \( 1 + (0.382 - 0.923i)T \)
13 \( 1 + (-0.649 - 0.760i)T \)
17 \( 1 + (-0.972 - 0.233i)T \)
19 \( 1 + (0.923 + 0.382i)T \)
23 \( 1 + (0.0784 - 0.996i)T \)
29 \( 1 + (0.987 + 0.156i)T \)
31 \( 1 + (0.233 - 0.972i)T \)
37 \( 1 + (0.649 - 0.760i)T \)
41 \( 1 + (-0.987 + 0.156i)T \)
43 \( 1 + (-0.852 - 0.522i)T \)
47 \( 1 + (-0.156 + 0.987i)T \)
53 \( 1 + (0.987 + 0.156i)T \)
59 \( 1 + (0.453 + 0.891i)T \)
61 \( 1 + (-0.453 + 0.891i)T \)
67 \( 1 + (-0.156 - 0.987i)T \)
71 \( 1 + (0.852 + 0.522i)T \)
73 \( 1 + (-0.649 + 0.760i)T \)
79 \( 1 + (-0.453 - 0.891i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (-0.382 - 0.923i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−21.511115786342380463663146144296, −20.47902477813019194832374446221, −19.778544206250468964442754247091, −19.60929503306558910177612984868, −18.21337030801691418251065622296, −17.46471689456443672215799939279, −16.96403241846266997558980218147, −16.154934215534100298431348167838, −15.33442133706045622330274245902, −14.813762472179626017795014787941, −13.95859660281782986405783357255, −13.5028159816557906470726706359, −11.78100410646370995194097619297, −11.14598275783854221517938197563, −10.0852584686442335864097278171, −9.769659768103295001519076161223, −8.86126012253051131704171388410, −8.092371297784957188461594335748, −7.18252849335449780572794455141, −6.61637047101451462216064858528, −5.00432244663001453486208126833, −4.78798309709625620270145023520, −3.73939566545626524937562684454, −2.30595187277046408233442326029, −1.3608552653265260192648626465, 0.51734871389135191127764093114, 1.558271901964780704760921911719, 2.59207196042047273488997367266, 2.999193926843634004989355659109, 4.30147800233261903790838236462, 5.57574721333732046788816533554, 6.56940401103977156459119697297, 7.52117897840054852564419199087, 8.309264067309444622020918297849, 8.76604772825821295499927415929, 9.60058987970676503011433778720, 10.67620705169991411711094045002, 11.668041728069533007118411644995, 11.992642224331286470589160882597, 12.90993571183910266500814784455, 13.63286401716708142748471799173, 14.50990372005222802013010621443, 15.40418626691322395812197123179, 16.42443534722150110399348252578, 17.33465345961162773119885816510, 18.06367159902200663605967844423, 18.492402858326158972467728370958, 19.25193272372860827461704800502, 19.99829749130779327425136562964, 20.51350403159221708368512907896

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