Properties

Degree $2$
Conductor $668$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.55·3-s + 2·5-s + 4.48·7-s − 0.591·9-s − 1.62·11-s + 3.38·13-s − 3.10·15-s − 5.30·17-s + 6.97·19-s − 6.96·21-s + 0.132·23-s − 25-s + 5.57·27-s − 7.86·29-s + 10.5·31-s + 2.52·33-s + 8.97·35-s + 1.72·37-s − 5.25·39-s + 8.40·41-s − 6.76·43-s − 1.18·45-s + 0.208·47-s + 13.1·49-s + 8.23·51-s + 8.68·53-s − 3.25·55-s + ⋯
L(s)  = 1  − 0.896·3-s + 0.894·5-s + 1.69·7-s − 0.197·9-s − 0.489·11-s + 0.938·13-s − 0.801·15-s − 1.28·17-s + 1.59·19-s − 1.51·21-s + 0.0276·23-s − 0.200·25-s + 1.07·27-s − 1.46·29-s + 1.90·31-s + 0.439·33-s + 1.51·35-s + 0.282·37-s − 0.840·39-s + 1.31·41-s − 1.03·43-s − 0.176·45-s + 0.0304·47-s + 1.87·49-s + 1.15·51-s + 1.19·53-s − 0.438·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(668\)    =    \(2^{2} \cdot 167\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{668} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 668,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51025\)
\(L(\frac12)\) \(\approx\) \(1.51025\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
167 \( 1 + T \)
good3 \( 1 + 1.55T + 3T^{2} \)
5 \( 1 - 2T + 5T^{2} \)
7 \( 1 - 4.48T + 7T^{2} \)
11 \( 1 + 1.62T + 11T^{2} \)
13 \( 1 - 3.38T + 13T^{2} \)
17 \( 1 + 5.30T + 17T^{2} \)
19 \( 1 - 6.97T + 19T^{2} \)
23 \( 1 - 0.132T + 23T^{2} \)
29 \( 1 + 7.86T + 29T^{2} \)
31 \( 1 - 10.5T + 31T^{2} \)
37 \( 1 - 1.72T + 37T^{2} \)
41 \( 1 - 8.40T + 41T^{2} \)
43 \( 1 + 6.76T + 43T^{2} \)
47 \( 1 - 0.208T + 47T^{2} \)
53 \( 1 - 8.68T + 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 + 3.06T + 61T^{2} \)
67 \( 1 - 7.51T + 67T^{2} \)
71 \( 1 - 2.06T + 71T^{2} \)
73 \( 1 - 5.43T + 73T^{2} \)
79 \( 1 + 17.5T + 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 + 4.21T + 97T^{2} \)
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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

−10.74253065789660543906904368787, −9.789295911241415290327031779344, −8.707667076922791242721586351521, −7.987144998003215799959593724236, −6.83501149067447713041548073367, −5.69832933100411819944606081136, −5.34030650694394925755197301763, −4.28328384991421060527371329937, −2.47267815614508293982749650367, −1.22144523336725543105635312858, 1.22144523336725543105635312858, 2.47267815614508293982749650367, 4.28328384991421060527371329937, 5.34030650694394925755197301763, 5.69832933100411819944606081136, 6.83501149067447713041548073367, 7.987144998003215799959593724236, 8.707667076922791242721586351521, 9.789295911241415290327031779344, 10.74253065789660543906904368787

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