Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 167 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2.31·5-s + 4.58·7-s + 9-s + 3.21·11-s − 5.90·13-s + 2.31·15-s + 6.31·17-s − 7.08·19-s + 4.58·21-s − 1.12·23-s + 0.350·25-s + 27-s − 0.0956·29-s + 4.81·31-s + 3.21·33-s + 10.6·35-s + 3.60·37-s − 5.90·39-s + 8.59·41-s − 8.44·43-s + 2.31·45-s + 6.42·47-s + 14.0·49-s + 6.31·51-s + 2.56·53-s + 7.43·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.03·5-s + 1.73·7-s + 0.333·9-s + 0.969·11-s − 1.63·13-s + 0.597·15-s + 1.53·17-s − 1.62·19-s + 1.00·21-s − 0.233·23-s + 0.0700·25-s + 0.192·27-s − 0.0177·29-s + 0.865·31-s + 0.559·33-s + 1.79·35-s + 0.593·37-s − 0.946·39-s + 1.34·41-s − 1.28·43-s + 0.344·45-s + 0.937·47-s + 2.00·49-s + 0.883·51-s + 0.351·53-s + 1.00·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4008} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.756798815$
$L(\frac12)$  $\approx$  $3.756798815$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;167\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;167\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
167 \( 1 - T \)
good5 \( 1 - 2.31T + 5T^{2} \)
7 \( 1 - 4.58T + 7T^{2} \)
11 \( 1 - 3.21T + 11T^{2} \)
13 \( 1 + 5.90T + 13T^{2} \)
17 \( 1 - 6.31T + 17T^{2} \)
19 \( 1 + 7.08T + 19T^{2} \)
23 \( 1 + 1.12T + 23T^{2} \)
29 \( 1 + 0.0956T + 29T^{2} \)
31 \( 1 - 4.81T + 31T^{2} \)
37 \( 1 - 3.60T + 37T^{2} \)
41 \( 1 - 8.59T + 41T^{2} \)
43 \( 1 + 8.44T + 43T^{2} \)
47 \( 1 - 6.42T + 47T^{2} \)
53 \( 1 - 2.56T + 53T^{2} \)
59 \( 1 - 6.13T + 59T^{2} \)
61 \( 1 + 5.66T + 61T^{2} \)
67 \( 1 + 9.24T + 67T^{2} \)
71 \( 1 + 7.95T + 71T^{2} \)
73 \( 1 - 14.9T + 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 + 3.20T + 83T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 - 0.476T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.352950683091227130244266366883, −7.85134142330747854946897961989, −7.12812590564455591180727575293, −6.18161237605486147527178935773, −5.40895877314317540235508295240, −4.65363707230646831700723110504, −4.02470775141396009402871524832, −2.62067657964600691705809484343, −2.01699940121795650077591486045, −1.21133205348591590088771980689, 1.21133205348591590088771980689, 2.01699940121795650077591486045, 2.62067657964600691705809484343, 4.02470775141396009402871524832, 4.65363707230646831700723110504, 5.40895877314317540235508295240, 6.18161237605486147527178935773, 7.12812590564455591180727575293, 7.85134142330747854946897961989, 8.352950683091227130244266366883

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