Properties

Degree 2
Conductor $ 2^{2} \cdot 19 \cdot 79 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.45·5-s − 5.04·7-s − 3·9-s − 2.96·11-s − 6.51·13-s − 6.41·17-s + 19-s + 7.57·23-s + 1.00·25-s + 9.36·29-s − 6.09·31-s − 12.3·35-s − 0.659·37-s + 4.13·41-s − 1.33·43-s − 7.35·45-s − 11.5·47-s + 18.4·49-s − 1.34·53-s − 7.27·55-s + 5.20·59-s + 8.44·61-s + 15.1·63-s − 15.9·65-s − 4.86·67-s + 2.09·71-s + 5.55·73-s + ⋯
L(s)  = 1  + 1.09·5-s − 1.90·7-s − 9-s − 0.895·11-s − 1.80·13-s − 1.55·17-s + 0.229·19-s + 1.58·23-s + 0.201·25-s + 1.73·29-s − 1.09·31-s − 2.08·35-s − 0.108·37-s + 0.645·41-s − 0.203·43-s − 1.09·45-s − 1.67·47-s + 2.63·49-s − 0.184·53-s − 0.981·55-s + 0.678·59-s + 1.08·61-s + 1.90·63-s − 1.98·65-s − 0.594·67-s + 0.248·71-s + 0.649·73-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6004\)    =    \(2^{2} \cdot 19 \cdot 79\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6004} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7223522926$
$L(\frac12)$  $\approx$  $0.7223522926$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;79\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;19,\;79\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
19 \( 1 - T \)
79 \( 1 - T \)
good3 \( 1 + 3T^{2} \)
5 \( 1 - 2.45T + 5T^{2} \)
7 \( 1 + 5.04T + 7T^{2} \)
11 \( 1 + 2.96T + 11T^{2} \)
13 \( 1 + 6.51T + 13T^{2} \)
17 \( 1 + 6.41T + 17T^{2} \)
23 \( 1 - 7.57T + 23T^{2} \)
29 \( 1 - 9.36T + 29T^{2} \)
31 \( 1 + 6.09T + 31T^{2} \)
37 \( 1 + 0.659T + 37T^{2} \)
41 \( 1 - 4.13T + 41T^{2} \)
43 \( 1 + 1.33T + 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 + 1.34T + 53T^{2} \)
59 \( 1 - 5.20T + 59T^{2} \)
61 \( 1 - 8.44T + 61T^{2} \)
67 \( 1 + 4.86T + 67T^{2} \)
71 \( 1 - 2.09T + 71T^{2} \)
73 \( 1 - 5.55T + 73T^{2} \)
83 \( 1 + 1.15T + 83T^{2} \)
89 \( 1 - 7.00T + 89T^{2} \)
97 \( 1 + 8.42T + 97T^{2} \)
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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

−8.167136186978467331820739534297, −6.94954312289246028149038333910, −6.82288892709415646066982542321, −5.94935483529131448402272422238, −5.29626959349422490394714362522, −4.67812131794134052987014824986, −3.29275946251221900391537076247, −2.67392106407307981181971101220, −2.28029592204728095805483094219, −0.40470571370775683434275271680, 0.40470571370775683434275271680, 2.28029592204728095805483094219, 2.67392106407307981181971101220, 3.29275946251221900391537076247, 4.67812131794134052987014824986, 5.29626959349422490394714362522, 5.94935483529131448402272422238, 6.82288892709415646066982542321, 6.94954312289246028149038333910, 8.167136186978467331820739534297

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