Properties

Label 2-668-668.667-c1-0-80
Degree $2$
Conductor $668$
Sign $0.00602 - 0.999i$
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  + (0.00426 − 1.41i)2-s − 3.24i·3-s + (−1.99 − 0.0120i)4-s + (−4.58 − 0.0138i)6-s − 4.80i·7-s + (−0.0255 + 2.82i)8-s − 7.53·9-s + 6.45i·11-s + (−0.0391 + 6.49i)12-s + (−6.79 − 0.0204i)14-s + (3.99 + 0.0482i)16-s + (−0.0321 + 10.6i)18-s − 7.38i·19-s − 15.6·21-s + (9.12 + 0.0275i)22-s + ⋯
L(s)  = 1  + (0.00301 − 0.999i)2-s − 1.87i·3-s + (−0.999 − 0.00602i)4-s + (−1.87 − 0.00564i)6-s − 1.81i·7-s + (−0.00904 + 0.999i)8-s − 2.51·9-s + 1.94i·11-s + (−0.0112 + 1.87i)12-s + (−1.81 − 0.00547i)14-s + (0.999 + 0.0120i)16-s + (−0.00756 + 2.51i)18-s − 1.69i·19-s − 3.40·21-s + (1.94 + 0.00586i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00602 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00602 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(668\)    =    \(2^{2} \cdot 167\)
Sign: $0.00602 - 0.999i$
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: \((2,\ 668,\ (\ :1/2),\ 0.00602 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.644640 + 0.640766i\)
\(L(\frac12)\) \(\approx\) \(0.644640 + 0.640766i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.00426 + 1.41i)T \)
167 \( 1 + 12.9iT \)
good3 \( 1 + 3.24iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + 4.80iT - 7T^{2} \)
11 \( 1 - 6.45iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 7.38iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 6.97T + 29T^{2} \)
31 \( 1 + 6.33iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 0.477iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 12.3T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 9.94T + 89T^{2} \)
97 \( 1 - 3.38T + 97T^{2} \)
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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

−9.993396572657542102061136449701, −9.081610169848078762624352856781, −7.85236815897992675326373981969, −7.25686737363009728944849725851, −6.72233189638883326115456473973, −5.09222566084158627252082265748, −4.07639307242490744381379921834, −2.61019064526994542503736731037, −1.61355347631888097058804336533, −0.49894444634149793867038996534, 3.07230023016038156614558227729, 3.77637968895080576047040399196, 5.16336974311861866860258354612, 5.60652490968909816637317836872, 6.20872558350290771067059042540, 8.200061373840130009498049654207, 8.664808327240319532353793950822, 9.172016924498629217935982281188, 10.05840531576385438864973222507, 10.96575763692372758616919164144

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