Properties

Label 2-8008-1.1-c1-0-41
Degree $2$
Conductor $8008$
Sign $1$
Analytic cond. $63.9442$
Root an. cond. $7.99651$
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  + 1.37·3-s − 0.949·5-s + 7-s − 1.11·9-s − 11-s + 13-s − 1.30·15-s − 5.27·17-s − 3.79·19-s + 1.37·21-s + 2.13·23-s − 4.09·25-s − 5.65·27-s + 2.06·29-s − 0.698·31-s − 1.37·33-s − 0.949·35-s + 5.10·37-s + 1.37·39-s − 1.72·41-s + 0.353·43-s + 1.05·45-s + 12.2·47-s + 49-s − 7.24·51-s + 12.3·53-s + 0.949·55-s + ⋯
L(s)  = 1  + 0.793·3-s − 0.424·5-s + 0.377·7-s − 0.370·9-s − 0.301·11-s + 0.277·13-s − 0.336·15-s − 1.27·17-s − 0.870·19-s + 0.299·21-s + 0.444·23-s − 0.819·25-s − 1.08·27-s + 0.384·29-s − 0.125·31-s − 0.239·33-s − 0.160·35-s + 0.839·37-s + 0.220·39-s − 0.269·41-s + 0.0539·43-s + 0.157·45-s + 1.78·47-s + 0.142·49-s − 1.01·51-s + 1.69·53-s + 0.128·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8008\)    =    \(2^{3} \cdot 7 \cdot 11 \cdot 13\)
Sign: $1$
Analytic conductor: \(63.9442\)
Root analytic conductor: \(7.99651\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8008,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.005778425\)
\(L(\frac12)\) \(\approx\) \(2.005778425\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 - T \)
good3 \( 1 - 1.37T + 3T^{2} \)
5 \( 1 + 0.949T + 5T^{2} \)
17 \( 1 + 5.27T + 17T^{2} \)
19 \( 1 + 3.79T + 19T^{2} \)
23 \( 1 - 2.13T + 23T^{2} \)
29 \( 1 - 2.06T + 29T^{2} \)
31 \( 1 + 0.698T + 31T^{2} \)
37 \( 1 - 5.10T + 37T^{2} \)
41 \( 1 + 1.72T + 41T^{2} \)
43 \( 1 - 0.353T + 43T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 - 12.3T + 53T^{2} \)
59 \( 1 + 0.154T + 59T^{2} \)
61 \( 1 - 5.42T + 61T^{2} \)
67 \( 1 - 1.39T + 67T^{2} \)
71 \( 1 - 3.90T + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 - 6.64T + 89T^{2} \)
97 \( 1 - 9.20T + 97T^{2} \)
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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

−8.024251377695937305420851057265, −7.28076690768276072564276295449, −6.52684825024526651781952790323, −5.77309219367989877838028403322, −4.93822659145617521618527271394, −4.10867144212817016210287402082, −3.61095894675585764311596158876, −2.46554655785580945253011285677, −2.15164517450064558272704458266, −0.64519794628026040364954156739, 0.64519794628026040364954156739, 2.15164517450064558272704458266, 2.46554655785580945253011285677, 3.61095894675585764311596158876, 4.10867144212817016210287402082, 4.93822659145617521618527271394, 5.77309219367989877838028403322, 6.52684825024526651781952790323, 7.28076690768276072564276295449, 8.024251377695937305420851057265

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