Properties

Label 2-8004-1.1-c1-0-25
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 + 3.56·5-s − 0.955·7-s + 9-s − 5.80·11-s + 2.78·13-s − 3.56·15-s − 4.80·17-s − 2.83·19-s + 0.955·21-s + 23-s + 7.70·25-s − 27-s + 29-s + 1.35·31-s + 5.80·33-s − 3.40·35-s − 2.67·37-s − 2.78·39-s + 10.9·41-s − 5.88·43-s + 3.56·45-s − 0.527·47-s − 6.08·49-s + 4.80·51-s − 7.00·53-s − 20.6·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.59·5-s − 0.361·7-s + 0.333·9-s − 1.74·11-s + 0.772·13-s − 0.920·15-s − 1.16·17-s − 0.650·19-s + 0.208·21-s + 0.208·23-s + 1.54·25-s − 0.192·27-s + 0.185·29-s + 0.243·31-s + 1.00·33-s − 0.575·35-s − 0.440·37-s − 0.446·39-s + 1.70·41-s − 0.898·43-s + 0.531·45-s − 0.0768·47-s − 0.869·49-s + 0.672·51-s − 0.962·53-s − 2.78·55-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: $\chi_{8004} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8004,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.711016509\)
\(L(\frac12)\) \(\approx\) \(1.711016509\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.56T + 5T^{2} \)
7 \( 1 + 0.955T + 7T^{2} \)
11 \( 1 + 5.80T + 11T^{2} \)
13 \( 1 - 2.78T + 13T^{2} \)
17 \( 1 + 4.80T + 17T^{2} \)
19 \( 1 + 2.83T + 19T^{2} \)
31 \( 1 - 1.35T + 31T^{2} \)
37 \( 1 + 2.67T + 37T^{2} \)
41 \( 1 - 10.9T + 41T^{2} \)
43 \( 1 + 5.88T + 43T^{2} \)
47 \( 1 + 0.527T + 47T^{2} \)
53 \( 1 + 7.00T + 53T^{2} \)
59 \( 1 - 2.89T + 59T^{2} \)
61 \( 1 - 8.42T + 61T^{2} \)
67 \( 1 - 11.0T + 67T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 - 5.58T + 73T^{2} \)
79 \( 1 + 9.15T + 79T^{2} \)
83 \( 1 - 5.98T + 83T^{2} \)
89 \( 1 - 8.05T + 89T^{2} \)
97 \( 1 - 8.87T + 97T^{2} \)
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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

−7.87082621819494647595874525369, −6.82286997447650352540192373691, −6.38107952843305243345687665622, −5.80337665983159578117329687113, −5.14882152955329984324085784073, −4.59673244935518747303715628863, −3.39718045933901772110902647261, −2.41406707019870538441677920794, −1.95767708142481878546483232284, −0.64384141394601771768198564137, 0.64384141394601771768198564137, 1.95767708142481878546483232284, 2.41406707019870538441677920794, 3.39718045933901772110902647261, 4.59673244935518747303715628863, 5.14882152955329984324085784073, 5.80337665983159578117329687113, 6.38107952843305243345687665622, 6.82286997447650352540192373691, 7.87082621819494647595874525369

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