Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 23 \cdot 29 $
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 − 4.29·5-s + 3.54·7-s + 9-s + 0.161·11-s + 2.42·13-s − 4.29·15-s + 3.98·17-s + 7.43·19-s + 3.54·21-s + 23-s + 13.4·25-s + 27-s − 29-s + 2.05·31-s + 0.161·33-s − 15.2·35-s − 5.65·37-s + 2.42·39-s − 7.47·41-s − 2.80·43-s − 4.29·45-s − 5.80·47-s + 5.56·49-s + 3.98·51-s + 5.67·53-s − 0.694·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.92·5-s + 1.33·7-s + 0.333·9-s + 0.0487·11-s + 0.671·13-s − 1.10·15-s + 0.967·17-s + 1.70·19-s + 0.773·21-s + 0.208·23-s + 2.69·25-s + 0.192·27-s − 0.185·29-s + 0.368·31-s + 0.0281·33-s − 2.57·35-s − 0.929·37-s + 0.387·39-s − 1.16·41-s − 0.427·43-s − 0.640·45-s − 0.847·47-s + 0.795·49-s + 0.558·51-s + 0.779·53-s − 0.0936·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8004\)    =    \(2^{2} \cdot 3 \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8004} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.438041654$
$L(\frac12)$  $\approx$  $2.438041654$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;23,\;29\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 + T \)
good5 \( 1 + 4.29T + 5T^{2} \)
7 \( 1 - 3.54T + 7T^{2} \)
11 \( 1 - 0.161T + 11T^{2} \)
13 \( 1 - 2.42T + 13T^{2} \)
17 \( 1 - 3.98T + 17T^{2} \)
19 \( 1 - 7.43T + 19T^{2} \)
31 \( 1 - 2.05T + 31T^{2} \)
37 \( 1 + 5.65T + 37T^{2} \)
41 \( 1 + 7.47T + 41T^{2} \)
43 \( 1 + 2.80T + 43T^{2} \)
47 \( 1 + 5.80T + 47T^{2} \)
53 \( 1 - 5.67T + 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 - 14.8T + 61T^{2} \)
67 \( 1 + 9.34T + 67T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 - 2.93T + 73T^{2} \)
79 \( 1 + 0.741T + 79T^{2} \)
83 \( 1 + 7.89T + 83T^{2} \)
89 \( 1 - 1.43T + 89T^{2} \)
97 \( 1 + 4.91T + 97T^{2} \)
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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

−7.85852023413524582172106042221, −7.36292390216303213594127721069, −6.86061332221874666202543531812, −5.44477523003857558708794154120, −4.97410138615199809093849258373, −4.13515590485801314073851647060, −3.52781636843424051280708153790, −2.99669422135208905483596977733, −1.58515281328143001412263580175, −0.821400598416738282498528385174, 0.821400598416738282498528385174, 1.58515281328143001412263580175, 2.99669422135208905483596977733, 3.52781636843424051280708153790, 4.13515590485801314073851647060, 4.97410138615199809093849258373, 5.44477523003857558708794154120, 6.86061332221874666202543531812, 7.36292390216303213594127721069, 7.85852023413524582172106042221

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