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Properties

Label 1-667-667.160-r0-0-0
Degree $1$
Conductor $667$
Sign $-0.547 - 0.837i$
Analytic cond. $3.09753$
Root an. cond. $3.09753$
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  + (0.974 − 0.222i)2-s + (0.433 − 0.900i)3-s + (0.900 − 0.433i)4-s + (−0.222 − 0.974i)5-s + (0.222 − 0.974i)6-s + (0.900 + 0.433i)7-s + (0.781 − 0.623i)8-s + (−0.623 − 0.781i)9-s + (−0.433 − 0.900i)10-s + (−0.781 − 0.623i)11-s − i·12-s + (−0.623 + 0.781i)13-s + (0.974 + 0.222i)14-s + (−0.974 − 0.222i)15-s + (0.623 − 0.781i)16-s − i·17-s + ⋯
L(s)  = 1  + (0.974 − 0.222i)2-s + (0.433 − 0.900i)3-s + (0.900 − 0.433i)4-s + (−0.222 − 0.974i)5-s + (0.222 − 0.974i)6-s + (0.900 + 0.433i)7-s + (0.781 − 0.623i)8-s + (−0.623 − 0.781i)9-s + (−0.433 − 0.900i)10-s + (−0.781 − 0.623i)11-s − i·12-s + (−0.623 + 0.781i)13-s + (0.974 + 0.222i)14-s + (−0.974 − 0.222i)15-s + (0.623 − 0.781i)16-s − i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.547 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.547 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $-0.547 - 0.837i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ -0.547 - 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.423991986 - 2.631860870i\)
\(L(\frac12)\) \(\approx\) \(1.423991986 - 2.631860870i\)
\(L(1)\) \(\approx\) \(1.681031760 - 1.283032164i\)
\(L(1)\) \(\approx\) \(1.681031760 - 1.283032164i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.974 - 0.222i)T \)
3 \( 1 + (0.433 - 0.900i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
7 \( 1 + (0.900 + 0.433i)T \)
11 \( 1 + (-0.781 - 0.623i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.433 + 0.900i)T \)
31 \( 1 + (0.974 - 0.222i)T \)
37 \( 1 + (-0.781 + 0.623i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.974 + 0.222i)T \)
47 \( 1 + (-0.781 - 0.623i)T \)
53 \( 1 + (0.222 + 0.974i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.433 + 0.900i)T \)
67 \( 1 + (0.623 + 0.781i)T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (0.974 + 0.222i)T \)
79 \( 1 + (0.781 - 0.623i)T \)
83 \( 1 + (0.900 - 0.433i)T \)
89 \( 1 + (-0.974 + 0.222i)T \)
97 \( 1 + (-0.433 - 0.900i)T \)
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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

−22.85386330649550327158958885737, −22.33057444438577739775136359654, −21.411264506674659213737584458362, −20.92403510240254452664416898948, −19.96585313973394773634587551825, −19.42163316347167972831056293785, −17.81234451710486228254588559708, −17.323128555710804626771386821, −16.058028975955457696522865739521, −15.25959413452463898973816625593, −14.91386690022184089839227196745, −14.14850755606835075980023111114, −13.35046246673006125326156832744, −12.22310612416650755447317074368, −11.0792055594375401990713176401, −10.684557477946245054956774666626, −9.843237044878515159328008337, −8.13294800453722236327725671305, −7.74268854424416392041805532418, −6.71131976026478075670137747273, −5.37601722062845710508091667370, −4.744092662266729889597888180768, −3.8152945761529234061698871850, −2.88791329495175212474514349490, −2.11810825180456908504863486072, 1.01391085928269515089566446601, 2.00506589007615199517512755177, 2.86130700668601963946332196425, 4.137854761502362671445473316533, 5.15008025100963056669756743674, 5.75285517892043453393418998216, 7.06849163869543531478959163760, 7.86587012097631131761134600005, 8.64796742899053846534612640246, 9.76885572615331623998127026343, 11.2043427579204984728685171906, 12.01095083849982918711610088087, 12.2936184747846982557632039072, 13.51660434066008807198676117530, 13.89389642120921089925985971020, 14.804967574518333724722249644215, 15.75210793168708771903802618896, 16.56844691506643696329753839967, 17.62859260120588590281799254361, 18.7567017737905025251347380162, 19.238004666527666884783483138214, 20.39668576385519770769356117064, 20.787913514860757469793702275936, 21.45030282021731904710029291160, 22.665876554833861443514426777808

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