Properties

Label 2-8004-1.1-c1-0-39
Degree $2$
Conductor $8004$
Sign $1$
Analytic cond. $63.9122$
Root an. cond. $7.99451$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 0.235·5-s + 1.32·7-s + 9-s + 1.55·11-s − 3.66·13-s + 0.235·15-s − 2.58·17-s − 1.45·19-s + 1.32·21-s − 23-s − 4.94·25-s + 27-s + 29-s + 6.25·31-s + 1.55·33-s + 0.311·35-s − 9.15·37-s − 3.66·39-s + 11.0·41-s + 9.13·43-s + 0.235·45-s + 7.96·47-s − 5.25·49-s − 2.58·51-s − 5.80·53-s + 0.367·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.105·5-s + 0.499·7-s + 0.333·9-s + 0.469·11-s − 1.01·13-s + 0.0608·15-s − 0.627·17-s − 0.333·19-s + 0.288·21-s − 0.208·23-s − 0.988·25-s + 0.192·27-s + 0.185·29-s + 1.12·31-s + 0.271·33-s + 0.0526·35-s − 1.50·37-s − 0.587·39-s + 1.73·41-s + 1.39·43-s + 0.0351·45-s + 1.16·47-s − 0.750·49-s − 0.362·51-s − 0.797·53-s + 0.0495·55-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

Degree: \(2\)
Conductor: \(8004\)    =    \(2^{2} \cdot 3 \cdot 23 \cdot 29\)
Sign: $1$
Analytic conductor: \(63.9122\)
Root analytic conductor: \(7.99451\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8004,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.634553667\)
\(L(\frac12)\) \(\approx\) \(2.634553667\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 - T \)
good5 \( 1 - 0.235T + 5T^{2} \)
7 \( 1 - 1.32T + 7T^{2} \)
11 \( 1 - 1.55T + 11T^{2} \)
13 \( 1 + 3.66T + 13T^{2} \)
17 \( 1 + 2.58T + 17T^{2} \)
19 \( 1 + 1.45T + 19T^{2} \)
31 \( 1 - 6.25T + 31T^{2} \)
37 \( 1 + 9.15T + 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 - 9.13T + 43T^{2} \)
47 \( 1 - 7.96T + 47T^{2} \)
53 \( 1 + 5.80T + 53T^{2} \)
59 \( 1 - 5.27T + 59T^{2} \)
61 \( 1 - 12.9T + 61T^{2} \)
67 \( 1 + 0.0629T + 67T^{2} \)
71 \( 1 - 1.10T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 - 12.5T + 79T^{2} \)
83 \( 1 - 11.0T + 83T^{2} \)
89 \( 1 - 9.93T + 89T^{2} \)
97 \( 1 + 0.659T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.922559735532104554314707682007, −7.20726031842460544111307676429, −6.56524989796469319329588569619, −5.76563673267363604751899689414, −4.89115725391632747879882280722, −4.28373735048288416703313255205, −3.57626664076770760413659262096, −2.42939398855383757434466282180, −2.05309755985006742815764141763, −0.77310971636436325791080797292, 0.77310971636436325791080797292, 2.05309755985006742815764141763, 2.42939398855383757434466282180, 3.57626664076770760413659262096, 4.28373735048288416703313255205, 4.89115725391632747879882280722, 5.76563673267363604751899689414, 6.56524989796469319329588569619, 7.20726031842460544111307676429, 7.922559735532104554314707682007

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