Properties

Label 2-668-668.667-c1-0-35
Degree $2$
Conductor $668$
Sign $0.751 - 0.659i$
Analytic cond. $5.33400$
Root an. cond. $2.30954$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.28 + 0.583i)2-s − 1.66i·3-s + (1.31 + 1.50i)4-s + (0.969 − 2.13i)6-s + 2.84i·7-s + (0.821 + 2.70i)8-s + 0.242·9-s + 5.38i·11-s + (2.49 − 2.18i)12-s + (−1.66 + 3.66i)14-s + (−0.521 + 3.96i)16-s + (0.312 + 0.141i)18-s − 5.77i·19-s + 4.72·21-s + (−3.14 + 6.94i)22-s + ⋯
L(s)  = 1  + (0.910 + 0.412i)2-s − 0.958i·3-s + (0.659 + 0.751i)4-s + (0.395 − 0.873i)6-s + 1.07i·7-s + (0.290 + 0.956i)8-s + 0.0808·9-s + 1.62i·11-s + (0.720 − 0.632i)12-s + (−0.444 + 0.980i)14-s + (−0.130 + 0.991i)16-s + (0.0736 + 0.0333i)18-s − 1.32i·19-s + 1.03·21-s + (−0.670 + 1.47i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(668\)    =    \(2^{2} \cdot 167\)
Sign: $0.751 - 0.659i$
Analytic conductor: \(5.33400\)
Root analytic conductor: \(2.30954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{668} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 668,\ (\ :1/2),\ 0.751 - 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.48345 + 0.934816i\)
\(L(\frac12)\) \(\approx\) \(2.48345 + 0.934816i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 - 0.583i)T \)
167 \( 1 - 12.9iT \)
good3 \( 1 + 1.66iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 - 2.84iT - 7T^{2} \)
11 \( 1 - 5.38iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 5.77iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 1.43T + 29T^{2} \)
31 \( 1 + 5.69iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 4.31iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 + 16.8T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.91890801471649885027465779597, −9.622250239961846772076009171490, −8.668159335937056252174674277906, −7.59362801095567040114681852001, −7.02283131688186023953195317919, −6.26062806768974372662438928838, −5.15672030388412105179565918614, −4.35197664073350861929312592585, −2.71045208601058045875129151499, −1.92609200096923150130703102698, 1.20866350310800929421114606735, 3.19055914892390262912693447112, 3.78402410710002792109086936784, 4.68654580494891356293122697500, 5.65521517643522385497610864171, 6.60962227878740515018728869265, 7.68525211243496167721405730512, 8.905142794253590281966904779786, 9.934515122710582246777735201797, 10.71104912941005139571455126432

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