Properties

Degree $2$
Conductor $668$
Sign $0.153 - 0.988i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.342 − 2.57i)3-s + (−0.628 − 0.277i)5-s + (−1.47 + 3.17i)7-s + (−3.60 + 0.977i)9-s + (−0.684 + 0.877i)11-s + (−1.75 + 3.42i)13-s + (−0.499 + 1.71i)15-s + (0.451 + 0.0342i)17-s + (−4.29 + 6.46i)19-s + (8.66 + 2.70i)21-s + (1.87 + 0.745i)23-s + (−3.04 − 3.34i)25-s + (0.736 + 1.75i)27-s + (−0.785 + 0.312i)29-s + (−1.72 + 0.399i)31-s + ⋯
L(s)  = 1  + (−0.197 − 1.48i)3-s + (−0.281 − 0.124i)5-s + (−0.557 + 1.19i)7-s + (−1.20 + 0.325i)9-s + (−0.206 + 0.264i)11-s + (−0.486 + 0.951i)13-s + (−0.128 + 0.442i)15-s + (0.109 + 0.00829i)17-s + (−0.985 + 1.48i)19-s + (1.88 + 0.590i)21-s + (0.391 + 0.155i)23-s + (−0.609 − 0.669i)25-s + (0.141 + 0.337i)27-s + (−0.145 + 0.0579i)29-s + (−0.310 + 0.0717i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(668\)    =    \(2^{2} \cdot 167\)
Sign: $0.153 - 0.988i$
Motivic weight: \(1\)
Character: $\chi_{668} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 668,\ (\ :1/2),\ 0.153 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.382252 + 0.327604i\)
\(L(\frac12)\) \(\approx\) \(0.382252 + 0.327604i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-9.42 - 8.84i)T \)
good3 \( 1 + (0.342 + 2.57i)T + (-2.89 + 0.785i)T^{2} \)
5 \( 1 + (0.628 + 0.277i)T + (3.36 + 3.69i)T^{2} \)
7 \( 1 + (1.47 - 3.17i)T + (-4.51 - 5.35i)T^{2} \)
11 \( 1 + (0.684 - 0.877i)T + (-2.67 - 10.6i)T^{2} \)
13 \( 1 + (1.75 - 3.42i)T + (-7.60 - 10.5i)T^{2} \)
17 \( 1 + (-0.451 - 0.0342i)T + (16.8 + 2.56i)T^{2} \)
19 \( 1 + (4.29 - 6.46i)T + (-7.35 - 17.5i)T^{2} \)
23 \( 1 + (-1.87 - 0.745i)T + (16.7 + 15.7i)T^{2} \)
29 \( 1 + (0.785 - 0.312i)T + (21.0 - 19.9i)T^{2} \)
31 \( 1 + (1.72 - 0.399i)T + (27.8 - 13.6i)T^{2} \)
37 \( 1 + (3.03 + 0.823i)T + (31.9 + 18.7i)T^{2} \)
41 \( 1 + (-8.80 + 4.29i)T + (25.2 - 32.3i)T^{2} \)
43 \( 1 + (-0.281 + 2.96i)T + (-42.2 - 8.08i)T^{2} \)
47 \( 1 + (-0.207 - 10.9i)T + (-46.9 + 1.77i)T^{2} \)
53 \( 1 + (-1.65 - 9.63i)T + (-49.9 + 17.7i)T^{2} \)
59 \( 1 + (11.8 - 0.896i)T + (58.3 - 8.89i)T^{2} \)
61 \( 1 + (-10.0 + 10.2i)T + (-1.15 - 60.9i)T^{2} \)
67 \( 1 + (4.03 - 1.78i)T + (45.0 - 49.5i)T^{2} \)
71 \( 1 + (8.69 - 0.991i)T + (69.1 - 15.9i)T^{2} \)
73 \( 1 + (1.19 - 0.892i)T + (20.4 - 70.0i)T^{2} \)
79 \( 1 + (2.45 - 6.52i)T + (-59.4 - 52.0i)T^{2} \)
83 \( 1 + (1.50 - 2.08i)T + (-26.2 - 78.7i)T^{2} \)
89 \( 1 + (-1.02 - 3.52i)T + (-75.0 + 47.8i)T^{2} \)
97 \( 1 + (2.71 + 0.626i)T + (87.1 + 42.5i)T^{2} \)
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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

−10.91572311642657431522000959958, −9.688002083817141455334094933098, −8.839624835183244346929901238078, −7.960310978547250267133865750941, −7.19686264152497770178414711092, −6.23783280290668653436406780190, −5.69840942568320793464410375452, −4.20476573167986679405300081107, −2.60812120002550306865056263708, −1.73476073887647222870780336434, 0.26343445350389957571545875738, 2.93582065916221670748103319820, 3.82712979452101853705565096509, 4.62851637507661783513273692604, 5.54553220004420202184129451878, 6.80252975825968121215898995417, 7.67020123245358281823304683681, 8.842365369270836071912073919252, 9.662703231992161313259058851882, 10.43015388734673305011175055264

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