Properties

Label 2-8004-1.1-c1-0-14
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 − 0.309·5-s − 1.15·7-s + 9-s + 5.92·11-s − 4.60·13-s + 0.309·15-s − 7.33·17-s − 3.96·19-s + 1.15·21-s + 23-s − 4.90·25-s − 27-s + 29-s + 7.27·31-s − 5.92·33-s + 0.358·35-s + 2.80·37-s + 4.60·39-s − 4.27·41-s + 5.11·43-s − 0.309·45-s + 3.04·47-s − 5.66·49-s + 7.33·51-s − 13.6·53-s − 1.83·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.138·5-s − 0.436·7-s + 0.333·9-s + 1.78·11-s − 1.27·13-s + 0.0800·15-s − 1.77·17-s − 0.909·19-s + 0.252·21-s + 0.208·23-s − 0.980·25-s − 0.192·27-s + 0.185·29-s + 1.30·31-s − 1.03·33-s + 0.0605·35-s + 0.460·37-s + 0.737·39-s − 0.667·41-s + 0.780·43-s − 0.0461·45-s + 0.443·47-s − 0.809·49-s + 1.02·51-s − 1.87·53-s − 0.247·55-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

\(L(1)\) \(\approx\) \(1.028433179\)
\(L(\frac12)\) \(\approx\) \(1.028433179\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.309T + 5T^{2} \)
7 \( 1 + 1.15T + 7T^{2} \)
11 \( 1 - 5.92T + 11T^{2} \)
13 \( 1 + 4.60T + 13T^{2} \)
17 \( 1 + 7.33T + 17T^{2} \)
19 \( 1 + 3.96T + 19T^{2} \)
31 \( 1 - 7.27T + 31T^{2} \)
37 \( 1 - 2.80T + 37T^{2} \)
41 \( 1 + 4.27T + 41T^{2} \)
43 \( 1 - 5.11T + 43T^{2} \)
47 \( 1 - 3.04T + 47T^{2} \)
53 \( 1 + 13.6T + 53T^{2} \)
59 \( 1 + 0.903T + 59T^{2} \)
61 \( 1 - 11.0T + 61T^{2} \)
67 \( 1 - 7.41T + 67T^{2} \)
71 \( 1 - 3.44T + 71T^{2} \)
73 \( 1 + 4.89T + 73T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 + 15.6T + 83T^{2} \)
89 \( 1 - 5.84T + 89T^{2} \)
97 \( 1 - 5.92T + 97T^{2} \)
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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

−7.76564240340225091653934756929, −6.73534810060401748764440920765, −6.65387344758839680537395205090, −5.97902970242888345462228564672, −4.82413089449410569251242826658, −4.39308536538691514731378100651, −3.71911977820810570872570728251, −2.55979519168282780603201805756, −1.77226875617993999202612478701, −0.50699869374103678000533268577, 0.50699869374103678000533268577, 1.77226875617993999202612478701, 2.55979519168282780603201805756, 3.71911977820810570872570728251, 4.39308536538691514731378100651, 4.82413089449410569251242826658, 5.97902970242888345462228564672, 6.65387344758839680537395205090, 6.73534810060401748764440920765, 7.76564240340225091653934756929

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