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Properties

Label 2-668-668.667-c1-0-49
Degree $2$
Conductor $668$
Sign $0.675 + 0.737i$
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  + (−1.37 − 0.317i)2-s + 2.42i·3-s + (1.79 + 0.876i)4-s − 2.41i·5-s + (0.771 − 3.34i)6-s + 1.33i·7-s + (−2.19 − 1.77i)8-s − 2.89·9-s + (−0.766 + 3.32i)10-s − 1.75i·11-s + (−2.12 + 4.36i)12-s − 3.38i·13-s + (0.423 − 1.83i)14-s + 5.85·15-s + (2.46 + 3.15i)16-s − 7.79i·17-s + ⋯
L(s)  = 1  + (−0.974 − 0.224i)2-s + 1.40i·3-s + (0.898 + 0.438i)4-s − 1.07i·5-s + (0.315 − 1.36i)6-s + 0.503i·7-s + (−0.777 − 0.628i)8-s − 0.963·9-s + (−0.242 + 1.05i)10-s − 0.528i·11-s + (−0.613 + 1.25i)12-s − 0.939i·13-s + (0.113 − 0.490i)14-s + 1.51·15-s + (0.616 + 0.787i)16-s − 1.89i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(668\)    =    \(2^{2} \cdot 167\)
Sign: $0.675 + 0.737i$
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: \((2,\ 668,\ (\ :1/2),\ 0.675 + 0.737i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.722800 - 0.318114i\)
\(L(\frac12)\) \(\approx\) \(0.722800 - 0.318114i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 + 0.317i)T \)
167 \( 1 + (12.0 + 4.74i)T \)
good3 \( 1 - 2.42iT - 3T^{2} \)
5 \( 1 + 2.41iT - 5T^{2} \)
7 \( 1 - 1.33iT - 7T^{2} \)
11 \( 1 + 1.75iT - 11T^{2} \)
13 \( 1 + 3.38iT - 13T^{2} \)
17 \( 1 + 7.79iT - 17T^{2} \)
19 \( 1 + 2.81iT - 19T^{2} \)
23 \( 1 + 1.74T + 23T^{2} \)
29 \( 1 + 4.79T + 29T^{2} \)
31 \( 1 + 4.24iT - 31T^{2} \)
37 \( 1 - 4.05iT - 37T^{2} \)
41 \( 1 - 1.69iT - 41T^{2} \)
43 \( 1 + 3.20T + 43T^{2} \)
47 \( 1 + 8.39iT - 47T^{2} \)
53 \( 1 + 4.10iT - 53T^{2} \)
59 \( 1 - 8.58T + 59T^{2} \)
61 \( 1 - 6.73T + 61T^{2} \)
67 \( 1 - 3.95T + 67T^{2} \)
71 \( 1 - 15.7T + 71T^{2} \)
73 \( 1 - 4.79iT - 73T^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 + 8.54T + 83T^{2} \)
89 \( 1 - 5.23T + 89T^{2} \)
97 \( 1 + 6.14T + 97T^{2} \)
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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

−10.03849933692709042448014959362, −9.603340859480564884056336731299, −8.837528560105178656565441051868, −8.298303373184119138388835916607, −7.09434874821493767872383537492, −5.58905635119337060115706243928, −4.98677216691318365392927857403, −3.68836990340284791428855769753, −2.60468272063481142542495625041, −0.59295074371520379660964195321, 1.49095235320840703298062074037, 2.27765470953302293544606345802, 3.81632660508412707044760991845, 5.85713349900873151402748376346, 6.60194765636177912440489534304, 7.12421492875002538575227032240, 7.81463580400583957979385751046, 8.641988278618637292544427641310, 9.856618523654150157716099707184, 10.59664395054426083465564390286

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