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Properties

Label 2-668-167.9-c1-0-8
Degree $2$
Conductor $668$
Sign $0.816 - 0.577i$
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  + (0.189 + 1.42i)3-s + (2.53 + 1.11i)5-s + (0.927 − 1.99i)7-s + (0.899 − 0.244i)9-s + (−1.13 + 1.45i)11-s + (2.39 − 4.67i)13-s + (−1.11 + 3.82i)15-s + (5.66 + 0.429i)17-s + (1.03 − 1.55i)19-s + (3.01 + 0.943i)21-s + (−3.88 − 1.54i)23-s + (1.78 + 1.96i)25-s + (2.18 + 5.21i)27-s + (−7.12 + 2.83i)29-s + (−2.93 + 0.679i)31-s + ⋯
L(s)  = 1  + (0.109 + 0.822i)3-s + (1.13 + 0.500i)5-s + (0.350 − 0.753i)7-s + (0.299 − 0.0813i)9-s + (−0.340 + 0.437i)11-s + (0.663 − 1.29i)13-s + (−0.287 + 0.986i)15-s + (1.37 + 0.104i)17-s + (0.237 − 0.357i)19-s + (0.658 + 0.205i)21-s + (−0.810 − 0.322i)23-s + (0.357 + 0.393i)25-s + (0.421 + 1.00i)27-s + (−1.32 + 0.526i)29-s + (−0.527 + 0.121i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(668\)    =    \(2^{2} \cdot 167\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(5.33400\)
Root analytic conductor: \(2.30954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{668} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 668,\ (\ :1/2),\ 0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96479 + 0.625176i\)
\(L(\frac12)\) \(\approx\) \(1.96479 + 0.625176i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.211 - 12.9i)T \)
good3 \( 1 + (-0.189 - 1.42i)T + (-2.89 + 0.785i)T^{2} \)
5 \( 1 + (-2.53 - 1.11i)T + (3.36 + 3.69i)T^{2} \)
7 \( 1 + (-0.927 + 1.99i)T + (-4.51 - 5.35i)T^{2} \)
11 \( 1 + (1.13 - 1.45i)T + (-2.67 - 10.6i)T^{2} \)
13 \( 1 + (-2.39 + 4.67i)T + (-7.60 - 10.5i)T^{2} \)
17 \( 1 + (-5.66 - 0.429i)T + (16.8 + 2.56i)T^{2} \)
19 \( 1 + (-1.03 + 1.55i)T + (-7.35 - 17.5i)T^{2} \)
23 \( 1 + (3.88 + 1.54i)T + (16.7 + 15.7i)T^{2} \)
29 \( 1 + (7.12 - 2.83i)T + (21.0 - 19.9i)T^{2} \)
31 \( 1 + (2.93 - 0.679i)T + (27.8 - 13.6i)T^{2} \)
37 \( 1 + (4.45 + 1.20i)T + (31.9 + 18.7i)T^{2} \)
41 \( 1 + (3.49 - 1.70i)T + (25.2 - 32.3i)T^{2} \)
43 \( 1 + (-0.277 + 2.92i)T + (-42.2 - 8.08i)T^{2} \)
47 \( 1 + (-0.140 - 7.44i)T + (-46.9 + 1.77i)T^{2} \)
53 \( 1 + (-2.20 - 12.8i)T + (-49.9 + 17.7i)T^{2} \)
59 \( 1 + (4.66 - 0.353i)T + (58.3 - 8.89i)T^{2} \)
61 \( 1 + (-7.59 + 7.74i)T + (-1.15 - 60.9i)T^{2} \)
67 \( 1 + (-4.57 + 2.02i)T + (45.0 - 49.5i)T^{2} \)
71 \( 1 + (13.5 - 1.54i)T + (69.1 - 15.9i)T^{2} \)
73 \( 1 + (-1.29 + 0.973i)T + (20.4 - 70.0i)T^{2} \)
79 \( 1 + (3.94 - 10.5i)T + (-59.4 - 52.0i)T^{2} \)
83 \( 1 + (-6.95 + 9.64i)T + (-26.2 - 78.7i)T^{2} \)
89 \( 1 + (2.28 + 7.84i)T + (-75.0 + 47.8i)T^{2} \)
97 \( 1 + (12.5 + 2.89i)T + (87.1 + 42.5i)T^{2} \)
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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

−10.43078144812861227354010360113, −9.979602867584057979566267816482, −9.166987786269446104538880031284, −7.904285552409083173067301035664, −7.16297873594212571367152944659, −5.89168590863547050906485027221, −5.22378259033876380986465481505, −3.98706339777329008316917665980, −3.04544300659858431847099896147, −1.48286600973642152625493253713, 1.50907665011178311153592926987, 2.10529432746228531000024188358, 3.78554361104431317935026263456, 5.33624585988616055710463116316, 5.80494265325948076037205723518, 6.84988518869484942497131734001, 7.87935277506372345224743249890, 8.669879184594327186386070118289, 9.533844347096407782940096853526, 10.22208262795675424427374969400

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