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 + 1.56·5-s + 3.21·7-s + 9-s − 5.96·11-s + 2.81·13-s + 1.56·15-s + 4.33·17-s + 4.82·19-s + 3.21·21-s + 3.36·23-s − 2.53·25-s + 27-s − 8.76·29-s + 2.34·31-s − 5.96·33-s + 5.04·35-s + 7.28·37-s + 2.81·39-s − 1.39·41-s + 2.86·43-s + 1.56·45-s + 9.87·47-s + 3.33·49-s + 4.33·51-s − 11.2·53-s − 9.36·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.701·5-s + 1.21·7-s + 0.333·9-s − 1.79·11-s + 0.780·13-s + 0.405·15-s + 1.05·17-s + 1.10·19-s + 0.701·21-s + 0.701·23-s − 0.507·25-s + 0.192·27-s − 1.62·29-s + 0.420·31-s − 1.03·33-s + 0.852·35-s + 1.19·37-s + 0.450·39-s − 0.217·41-s + 0.436·43-s + 0.233·45-s + 1.44·47-s + 0.476·49-s + 0.606·51-s − 1.54·53-s − 1.26·55-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  =  $(2,\ 4008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.253683918$
$L(\frac12)$  $\approx$  $3.253683918$
$L(\frac{3}{2})$   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 \( 1 - 1.56T + 5T^{2} \)
7 \( 1 - 3.21T + 7T^{2} \)
11 \( 1 + 5.96T + 11T^{2} \)
13 \( 1 - 2.81T + 13T^{2} \)
17 \( 1 - 4.33T + 17T^{2} \)
19 \( 1 - 4.82T + 19T^{2} \)
23 \( 1 - 3.36T + 23T^{2} \)
29 \( 1 + 8.76T + 29T^{2} \)
31 \( 1 - 2.34T + 31T^{2} \)
37 \( 1 - 7.28T + 37T^{2} \)
41 \( 1 + 1.39T + 41T^{2} \)
43 \( 1 - 2.86T + 43T^{2} \)
47 \( 1 - 9.87T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + 1.96T + 59T^{2} \)
61 \( 1 + 3.76T + 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 + 4.01T + 71T^{2} \)
73 \( 1 - 8.49T + 73T^{2} \)
79 \( 1 - 3.69T + 79T^{2} \)
83 \( 1 - 8.33T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 - 0.582T + 97T^{2} \)
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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

−8.154309615911745658731182065563, −7.88100091539284437933330570284, −7.33053770087076136439633542464, −6.01221438449615651787140399488, −5.39928051441401650473675333175, −4.89108630213649528125904139292, −3.73121959553667097196718800921, −2.83628911140750317993174786327, −2.02180665267423109110864251405, −1.08246060973286994053641051517, 1.08246060973286994053641051517, 2.02180665267423109110864251405, 2.83628911140750317993174786327, 3.73121959553667097196718800921, 4.89108630213649528125904139292, 5.39928051441401650473675333175, 6.01221438449615651787140399488, 7.33053770087076136439633542464, 7.88100091539284437933330570284, 8.154309615911745658731182065563

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