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Properties

Label 1-1617-1617.794-r1-0-0
Degree $1$
Conductor $1617$
Sign $0.500 + 0.865i$
Analytic cond. $173.770$
Root an. cond. $173.770$
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  + (−0.599 − 0.800i)2-s + (−0.280 + 0.959i)4-s + (0.712 − 0.701i)5-s + (0.936 − 0.351i)8-s + (−0.988 − 0.149i)10-s + (−0.393 + 0.919i)13-s + (−0.842 − 0.538i)16-s + (0.646 + 0.762i)17-s + (0.669 + 0.743i)19-s + (0.473 + 0.880i)20-s + (−0.0747 + 0.997i)23-s + (0.0149 − 0.999i)25-s + (0.971 − 0.237i)26-s + (0.134 − 0.990i)29-s + (0.104 − 0.994i)31-s + (0.0747 + 0.997i)32-s + ⋯
L(s)  = 1  + (−0.599 − 0.800i)2-s + (−0.280 + 0.959i)4-s + (0.712 − 0.701i)5-s + (0.936 − 0.351i)8-s + (−0.988 − 0.149i)10-s + (−0.393 + 0.919i)13-s + (−0.842 − 0.538i)16-s + (0.646 + 0.762i)17-s + (0.669 + 0.743i)19-s + (0.473 + 0.880i)20-s + (−0.0747 + 0.997i)23-s + (0.0149 − 0.999i)25-s + (0.971 − 0.237i)26-s + (0.134 − 0.990i)29-s + (0.104 − 0.994i)31-s + (0.0747 + 0.997i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.500 + 0.865i$
Analytic conductor: \(173.770\)
Root analytic conductor: \(173.770\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (794, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1617,\ (1:\ ),\ 0.500 + 0.865i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9956160684 + 0.5744115252i\)
\(L(\frac12)\) \(\approx\) \(0.9956160684 + 0.5744115252i\)
\(L(1)\) \(\approx\) \(0.8287831254 - 0.1976293379i\)
\(L(1)\) \(\approx\) \(0.8287831254 - 0.1976293379i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.599 - 0.800i)T \)
5 \( 1 + (0.712 - 0.701i)T \)
13 \( 1 + (-0.393 + 0.919i)T \)
17 \( 1 + (0.646 + 0.762i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (-0.0747 + 0.997i)T \)
29 \( 1 + (0.134 - 0.990i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (0.791 + 0.611i)T \)
41 \( 1 + (-0.936 + 0.351i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (0.575 - 0.817i)T \)
53 \( 1 + (0.337 + 0.941i)T \)
59 \( 1 + (-0.163 - 0.986i)T \)
61 \( 1 + (-0.337 + 0.941i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.473 + 0.880i)T \)
73 \( 1 + (0.575 + 0.817i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (0.393 + 0.919i)T \)
89 \( 1 + (-0.733 - 0.680i)T \)
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

−20.0203572767834943158829641253, −19.21151670039789068805819279701, −18.203913021148824276121528206208, −18.11833989057864013582714595024, −17.2276766420335720742260463112, −16.47275359110820885337129920307, −15.707747209939781119864543560073, −14.900595971902877253915999818677, −14.29592978266926598574918706777, −13.68136518290879819968926185668, −12.771864620526761936382644165441, −11.660725207469903046930777097889, −10.567241541742416361447919919229, −10.26961935784402596063446016175, −9.36361493579843378769516236126, −8.68240926700805169786162442707, −7.578668776043456146912646893480, −7.08173622182311317696636923021, −6.25874536892358804870576807473, −5.37387426129340993601932206339, −4.85264911735655791014064678459, −3.28235858576627623322052410328, −2.47086992178003386440244412668, −1.29338209343007266128293775472, −0.28083442519870677549194735917, 1.07598888195189516929919114940, 1.68986095793553039495991927953, 2.59020755742711469952746588805, 3.720722809100706137322417932954, 4.48125991287951287133250399939, 5.48419368868204379453568330030, 6.39239264679808108886344400775, 7.59111107758215479991320365591, 8.20262268456832964461732349359, 9.11669727172071504282778371351, 9.828027115347511947048293510998, 10.147222927661073036407276119556, 11.4862103041971830585028715542, 11.89791969226529570413820451307, 12.75571971401605122763358726371, 13.49721104396318461156722269085, 14.071444630933547411806580366459, 15.20769594542292481660627511036, 16.3208300188218034642469211828, 16.922367984582880845650655989886, 17.281259383956452595596957933891, 18.35255810096983696817612471074, 18.8180814229623108654535839782, 19.76560139852397398664061406906, 20.298154606991865593843084375064

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