Properties

Degree 2
Conductor $ 2^{3} \cdot 751 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.01·3-s − 4.06·5-s − 4.33·7-s + 1.04·9-s − 3.72·11-s − 4.52·13-s + 8.17·15-s − 5.00·17-s + 2.88·19-s + 8.71·21-s + 3.74·23-s + 11.5·25-s + 3.92·27-s + 9.42·29-s − 6.53·31-s + 7.48·33-s + 17.6·35-s − 6.59·37-s + 9.10·39-s − 2.05·41-s + 7.35·43-s − 4.26·45-s − 2.80·47-s + 11.7·49-s + 10.0·51-s + 11.5·53-s + 15.1·55-s + ⋯
L(s)  = 1  − 1.16·3-s − 1.81·5-s − 1.63·7-s + 0.349·9-s − 1.12·11-s − 1.25·13-s + 2.11·15-s − 1.21·17-s + 0.661·19-s + 1.90·21-s + 0.779·23-s + 2.30·25-s + 0.755·27-s + 1.75·29-s − 1.17·31-s + 1.30·33-s + 2.97·35-s − 1.08·37-s + 1.45·39-s − 0.321·41-s + 1.12·43-s − 0.635·45-s − 0.409·47-s + 1.67·49-s + 1.40·51-s + 1.58·53-s + 2.03·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6008\)    =    \(2^{3} \cdot 751\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6008} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6008,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;751\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;751\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 + T \)
good3 \( 1 + 2.01T + 3T^{2} \)
5 \( 1 + 4.06T + 5T^{2} \)
7 \( 1 + 4.33T + 7T^{2} \)
11 \( 1 + 3.72T + 11T^{2} \)
13 \( 1 + 4.52T + 13T^{2} \)
17 \( 1 + 5.00T + 17T^{2} \)
19 \( 1 - 2.88T + 19T^{2} \)
23 \( 1 - 3.74T + 23T^{2} \)
29 \( 1 - 9.42T + 29T^{2} \)
31 \( 1 + 6.53T + 31T^{2} \)
37 \( 1 + 6.59T + 37T^{2} \)
41 \( 1 + 2.05T + 41T^{2} \)
43 \( 1 - 7.35T + 43T^{2} \)
47 \( 1 + 2.80T + 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 + 13.0T + 59T^{2} \)
61 \( 1 - 4.80T + 61T^{2} \)
67 \( 1 - 1.53T + 67T^{2} \)
71 \( 1 - 8.02T + 71T^{2} \)
73 \( 1 + 16.0T + 73T^{2} \)
79 \( 1 + 7.19T + 79T^{2} \)
83 \( 1 + 13.5T + 83T^{2} \)
89 \( 1 - 3.29T + 89T^{2} \)
97 \( 1 - 5.35T + 97T^{2} \)
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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

−7.30250746057203341387582216751, −7.16641035370010404306862785435, −6.45310676449487197675006517831, −5.47843610148957256812846762899, −4.85749996329290306413575128611, −4.19163464344384105138333060828, −3.14505290865627274379591555553, −2.71852553385492893102236130313, −0.56861644557585869996959433509, 0, 0.56861644557585869996959433509, 2.71852553385492893102236130313, 3.14505290865627274379591555553, 4.19163464344384105138333060828, 4.85749996329290306413575128611, 5.47843610148957256812846762899, 6.45310676449487197675006517831, 7.16641035370010404306862785435, 7.30250746057203341387582216751

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