Properties

Label 2-4008-1.1-c1-0-31
Degree $2$
Conductor $4008$
Sign $1$
Analytic cond. $32.0040$
Root an. cond. $5.65721$
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 + 3.83·5-s + 4.48·7-s + 9-s + 1.10·11-s − 5.67·13-s − 3.83·15-s − 0.581·17-s − 4.72·19-s − 4.48·21-s − 3.33·23-s + 9.72·25-s − 27-s + 3.38·29-s + 2.78·31-s − 1.10·33-s + 17.1·35-s + 4.43·37-s + 5.67·39-s + 5.90·41-s + 10.0·43-s + 3.83·45-s − 12.4·47-s + 13.0·49-s + 0.581·51-s + 5.10·53-s + 4.24·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.71·5-s + 1.69·7-s + 0.333·9-s + 0.333·11-s − 1.57·13-s − 0.990·15-s − 0.140·17-s − 1.08·19-s − 0.978·21-s − 0.695·23-s + 1.94·25-s − 0.192·27-s + 0.628·29-s + 0.499·31-s − 0.192·33-s + 2.90·35-s + 0.728·37-s + 0.908·39-s + 0.921·41-s + 1.52·43-s + 0.571·45-s − 1.80·47-s + 1.86·49-s + 0.0813·51-s + 0.701·53-s + 0.572·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $1$
Analytic conductor: \(32.0040\)
Root analytic conductor: \(5.65721\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4008,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.658387707\)
\(L(\frac12)\) \(\approx\) \(2.658387707\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
167 \( 1 - T \)
good5 \( 1 - 3.83T + 5T^{2} \)
7 \( 1 - 4.48T + 7T^{2} \)
11 \( 1 - 1.10T + 11T^{2} \)
13 \( 1 + 5.67T + 13T^{2} \)
17 \( 1 + 0.581T + 17T^{2} \)
19 \( 1 + 4.72T + 19T^{2} \)
23 \( 1 + 3.33T + 23T^{2} \)
29 \( 1 - 3.38T + 29T^{2} \)
31 \( 1 - 2.78T + 31T^{2} \)
37 \( 1 - 4.43T + 37T^{2} \)
41 \( 1 - 5.90T + 41T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 + 12.4T + 47T^{2} \)
53 \( 1 - 5.10T + 53T^{2} \)
59 \( 1 + 2.77T + 59T^{2} \)
61 \( 1 - 9.42T + 61T^{2} \)
67 \( 1 - 9.07T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 - 9.14T + 73T^{2} \)
79 \( 1 + 17.4T + 79T^{2} \)
83 \( 1 - 16.9T + 83T^{2} \)
89 \( 1 + 1.50T + 89T^{2} \)
97 \( 1 - 0.761T + 97T^{2} \)
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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

−8.411365903584809591239992202030, −7.75131835505251994513454659138, −6.80050160965703137238173038571, −6.17972198001819775405518719170, −5.38972281261298235458676509977, −4.86127833213576516801858907472, −4.24210968510604448563935254663, −2.37591200970507392201063158374, −2.11011053981552507241462319258, −1.01229107372965360060543272958, 1.01229107372965360060543272958, 2.11011053981552507241462319258, 2.37591200970507392201063158374, 4.24210968510604448563935254663, 4.86127833213576516801858907472, 5.38972281261298235458676509977, 6.17972198001819775405518719170, 6.80050160965703137238173038571, 7.75131835505251994513454659138, 8.411365903584809591239992202030

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