Properties

Degree $2$
Conductor $668$
Sign $-0.944 + 0.327i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.353 + 2.65i)3-s + (−1.27 − 0.562i)5-s + (−1.92 + 4.13i)7-s + (−4.02 + 1.09i)9-s + (−1.02 + 1.30i)11-s + (1.42 − 2.77i)13-s + (1.04 − 3.57i)15-s + (−2.44 − 0.185i)17-s + (2.03 − 3.06i)19-s + (−11.6 − 3.64i)21-s + (0.849 + 0.337i)23-s + (−2.06 − 2.26i)25-s + (−1.21 − 2.88i)27-s + (−4.52 + 1.79i)29-s + (−2.76 + 0.638i)31-s + ⋯
L(s)  = 1  + (0.204 + 1.53i)3-s + (−0.568 − 0.251i)5-s + (−0.727 + 1.56i)7-s + (−1.34 + 0.363i)9-s + (−0.307 + 0.394i)11-s + (0.394 − 0.769i)13-s + (0.269 − 0.923i)15-s + (−0.591 − 0.0448i)17-s + (0.467 − 0.703i)19-s + (−2.54 − 0.795i)21-s + (0.177 + 0.0704i)23-s + (−0.412 − 0.453i)25-s + (−0.232 − 0.554i)27-s + (−0.840 + 0.334i)29-s + (−0.496 + 0.114i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(668\)    =    \(2^{2} \cdot 167\)
Sign: $-0.944 + 0.327i$
Motivic weight: \(1\)
Character: $\chi_{668} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 668,\ (\ :1/2),\ -0.944 + 0.327i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128923 - 0.764422i\)
\(L(\frac12)\) \(\approx\) \(0.128923 - 0.764422i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.39 + 12.4i)T \)
good3 \( 1 + (-0.353 - 2.65i)T + (-2.89 + 0.785i)T^{2} \)
5 \( 1 + (1.27 + 0.562i)T + (3.36 + 3.69i)T^{2} \)
7 \( 1 + (1.92 - 4.13i)T + (-4.51 - 5.35i)T^{2} \)
11 \( 1 + (1.02 - 1.30i)T + (-2.67 - 10.6i)T^{2} \)
13 \( 1 + (-1.42 + 2.77i)T + (-7.60 - 10.5i)T^{2} \)
17 \( 1 + (2.44 + 0.185i)T + (16.8 + 2.56i)T^{2} \)
19 \( 1 + (-2.03 + 3.06i)T + (-7.35 - 17.5i)T^{2} \)
23 \( 1 + (-0.849 - 0.337i)T + (16.7 + 15.7i)T^{2} \)
29 \( 1 + (4.52 - 1.79i)T + (21.0 - 19.9i)T^{2} \)
31 \( 1 + (2.76 - 0.638i)T + (27.8 - 13.6i)T^{2} \)
37 \( 1 + (-8.57 - 2.32i)T + (31.9 + 18.7i)T^{2} \)
41 \( 1 + (5.33 - 2.60i)T + (25.2 - 32.3i)T^{2} \)
43 \( 1 + (0.373 - 3.93i)T + (-42.2 - 8.08i)T^{2} \)
47 \( 1 + (-0.147 - 7.77i)T + (-46.9 + 1.77i)T^{2} \)
53 \( 1 + (-0.188 - 1.09i)T + (-49.9 + 17.7i)T^{2} \)
59 \( 1 + (10.3 - 0.787i)T + (58.3 - 8.89i)T^{2} \)
61 \( 1 + (-5.52 + 5.62i)T + (-1.15 - 60.9i)T^{2} \)
67 \( 1 + (8.24 - 3.64i)T + (45.0 - 49.5i)T^{2} \)
71 \( 1 + (-11.0 + 1.26i)T + (69.1 - 15.9i)T^{2} \)
73 \( 1 + (3.43 - 2.57i)T + (20.4 - 70.0i)T^{2} \)
79 \( 1 + (0.402 - 1.07i)T + (-59.4 - 52.0i)T^{2} \)
83 \( 1 + (4.94 - 6.85i)T + (-26.2 - 78.7i)T^{2} \)
89 \( 1 + (-0.517 - 1.77i)T + (-75.0 + 47.8i)T^{2} \)
97 \( 1 + (-5.32 - 1.23i)T + (87.1 + 42.5i)T^{2} \)
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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

−10.96394489646287876905284030309, −9.903856050891183409064801403628, −9.318436706780190325746033832241, −8.713374684610447801858853583730, −7.78759832362716346368103355643, −6.27715058462937473062998382819, −5.34982432247604290189117662256, −4.59039099204919690791787378378, −3.44018996270000832392683032614, −2.63420515633825528568827593987, 0.39245231931398659170018600367, 1.77577647522547028000445723807, 3.30939200346708601388437590918, 4.12464454773029808759940863601, 5.89462007111709987957596139503, 6.77795245524067419309008709015, 7.35725754699816963428087211051, 7.905615955379542964210641575746, 9.009596339757509572322760584677, 10.12201587184457130743627867399

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