Properties

Label 2-668-167.21-c1-0-3
Degree $2$
Conductor $668$
Sign $0.760 - 0.648i$
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  + (−1.77 − 0.270i)3-s + (−1.21 − 1.44i)5-s + (−0.212 + 0.294i)7-s + (0.209 + 0.0653i)9-s + (−2.42 + 2.29i)11-s + (−0.424 + 3.18i)13-s + (1.76 + 2.89i)15-s + (2.82 − 4.25i)17-s + (5.24 + 1.21i)19-s + (0.456 − 0.465i)21-s + (0.693 + 3.28i)23-s + (0.242 − 1.41i)25-s + (4.48 + 2.18i)27-s + (−0.511 + 2.42i)29-s + (0.832 − 0.159i)31-s + ⋯
L(s)  = 1  + (−1.02 − 0.156i)3-s + (−0.544 − 0.645i)5-s + (−0.0802 + 0.111i)7-s + (0.0697 + 0.0217i)9-s + (−0.732 + 0.691i)11-s + (−0.117 + 0.883i)13-s + (0.456 + 0.746i)15-s + (0.686 − 1.03i)17-s + (1.20 + 0.277i)19-s + (0.0996 − 0.101i)21-s + (0.144 + 0.684i)23-s + (0.0485 − 0.282i)25-s + (0.862 + 0.421i)27-s + (−0.0950 + 0.449i)29-s + (0.149 − 0.0286i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.760 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(668\)    =    \(2^{2} \cdot 167\)
Sign: $0.760 - 0.648i$
Analytic conductor: \(5.33400\)
Root analytic conductor: \(2.30954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{668} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 668,\ (\ :1/2),\ 0.760 - 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.672281 + 0.247749i\)
\(L(\frac12)\) \(\approx\) \(0.672281 + 0.247749i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.09 - 11.8i)T \)
good3 \( 1 + (1.77 + 0.270i)T + (2.86 + 0.894i)T^{2} \)
5 \( 1 + (1.21 + 1.44i)T + (-0.847 + 4.92i)T^{2} \)
7 \( 1 + (0.212 - 0.294i)T + (-2.21 - 6.64i)T^{2} \)
11 \( 1 + (2.42 - 2.29i)T + (0.624 - 10.9i)T^{2} \)
13 \( 1 + (0.424 - 3.18i)T + (-12.5 - 3.40i)T^{2} \)
17 \( 1 + (-2.82 + 4.25i)T + (-6.57 - 15.6i)T^{2} \)
19 \( 1 + (-5.24 - 1.21i)T + (17.0 + 8.33i)T^{2} \)
23 \( 1 + (-0.693 - 3.28i)T + (-21.0 + 9.30i)T^{2} \)
29 \( 1 + (0.511 - 2.42i)T + (-26.5 - 11.7i)T^{2} \)
31 \( 1 + (-0.832 + 0.159i)T + (28.8 - 11.4i)T^{2} \)
37 \( 1 + (0.474 - 0.148i)T + (30.4 - 21.0i)T^{2} \)
41 \( 1 + (0.439 - 0.174i)T + (29.8 - 28.1i)T^{2} \)
43 \( 1 + (-11.5 - 4.09i)T + (33.4 + 27.0i)T^{2} \)
47 \( 1 + (2.37 - 9.44i)T + (-41.4 - 22.2i)T^{2} \)
53 \( 1 + (-1.38 - 1.03i)T + (14.8 + 50.8i)T^{2} \)
59 \( 1 + (-2.10 - 3.16i)T + (-22.8 + 54.4i)T^{2} \)
61 \( 1 + (-4.62 - 5.92i)T + (-14.8 + 59.1i)T^{2} \)
67 \( 1 + (8.41 - 9.98i)T + (-11.3 - 66.0i)T^{2} \)
71 \( 1 + (1.33 + 14.0i)T + (-69.7 + 13.3i)T^{2} \)
73 \( 1 + (-3.57 + 6.37i)T + (-38.0 - 62.2i)T^{2} \)
79 \( 1 + (-7.02 - 0.266i)T + (78.7 + 5.97i)T^{2} \)
83 \( 1 + (7.61 - 2.06i)T + (71.6 - 41.9i)T^{2} \)
89 \( 1 + (-2.52 + 4.12i)T + (-40.5 - 79.2i)T^{2} \)
97 \( 1 + (17.4 + 3.33i)T + (90.1 + 35.8i)T^{2} \)
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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

−10.78594571405866334491157697743, −9.712793067431589115478112328625, −9.029971651416682071851244751670, −7.74792524217649630600681532585, −7.19940674018487709279105861190, −5.95947969569243508465306652741, −5.16426636027562282934321099129, −4.41875532861993038877011909856, −2.89378492421699604575153146008, −1.05964178159425784527457586783, 0.55119989187234276212014947519, 2.81405174075092636442255082124, 3.76995362574115881959837558525, 5.25493501095141633079478029174, 5.71179931192658189015875813367, 6.81527838821916330460463393858, 7.75940867247379808670665789662, 8.494362703248547713289520637116, 9.931522423825278110577733186790, 10.62139002024794452035449974879

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