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Properties

Label 2-668-668.667-c1-0-19
Degree $2$
Conductor $668$
Sign $0.545 - 0.837i$
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  + (−0.402 + 1.35i)2-s − 2.44i·3-s + (−1.67 − 1.09i)4-s + (3.32 + 0.986i)6-s + 2.87i·7-s + (2.15 − 1.83i)8-s − 3.00·9-s + 4.59i·11-s + (−2.67 + 4.10i)12-s + (−3.89 − 1.15i)14-s + (1.61 + 3.65i)16-s + (1.20 − 4.06i)18-s + 1.33i·19-s + 7.04·21-s + (−6.23 − 1.84i)22-s + ⋯
L(s)  = 1  + (−0.284 + 0.958i)2-s − 1.41i·3-s + (−0.837 − 0.545i)4-s + (1.35 + 0.402i)6-s + 1.08i·7-s + (0.761 − 0.648i)8-s − 1.00·9-s + 1.38i·11-s + (−0.771 + 1.18i)12-s + (−1.04 − 0.309i)14-s + (0.404 + 0.914i)16-s + (0.284 − 0.959i)18-s + 0.305i·19-s + 1.53·21-s + (−1.32 − 0.394i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(668\)    =    \(2^{2} \cdot 167\)
Sign: $0.545 - 0.837i$
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: \((2,\ 668,\ (\ :1/2),\ 0.545 - 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.988467 + 0.535883i\)
\(L(\frac12)\) \(\approx\) \(0.988467 + 0.535883i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.402 - 1.35i)T \)
167 \( 1 + 12.9iT \)
good3 \( 1 + 2.44iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 - 2.87iT - 7T^{2} \)
11 \( 1 - 4.59iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 1.33iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 9.11T + 29T^{2} \)
31 \( 1 - 3.49iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 10.6iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 18.7T + 89T^{2} \)
97 \( 1 + 7.64T + 97T^{2} \)
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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

−10.43840631860190949924827974056, −9.511089373169218207661622215190, −8.601021174368006014680640605647, −7.952213225566987393179153542270, −7.00216564068153237484345638191, −6.53235949631588310121974791053, −5.51756338871948239497086286818, −4.54641385085875049674312466209, −2.55819418876123053631731123653, −1.32357710696951243424510919038, 0.77821350374169742874949515137, 2.89545015072948542208708310431, 3.74012959733220578552855612215, 4.47814957614731417668104447787, 5.41753623934811615930362759357, 6.92836717336752169615835395164, 8.279676776419450731579697942272, 8.821590221097119383609403475969, 9.854835433443885382220981232520, 10.37159312806446806493311479671

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