Properties

Label 2-8007-1.1-c1-0-410
Degree $2$
Conductor $8007$
Sign $-1$
Analytic cond. $63.9362$
Root an. cond. $7.99601$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.09·2-s − 3-s + 2.38·4-s + 2.02·5-s − 2.09·6-s + 4.56·7-s + 0.804·8-s + 9-s + 4.24·10-s − 5.41·11-s − 2.38·12-s − 5.15·13-s + 9.55·14-s − 2.02·15-s − 3.08·16-s + 17-s + 2.09·18-s − 3.27·19-s + 4.83·20-s − 4.56·21-s − 11.3·22-s − 2.33·23-s − 0.804·24-s − 0.887·25-s − 10.8·26-s − 27-s + 10.8·28-s + ⋯
L(s)  = 1  + 1.48·2-s − 0.577·3-s + 1.19·4-s + 0.906·5-s − 0.854·6-s + 1.72·7-s + 0.284·8-s + 0.333·9-s + 1.34·10-s − 1.63·11-s − 0.688·12-s − 1.43·13-s + 2.55·14-s − 0.523·15-s − 0.771·16-s + 0.242·17-s + 0.493·18-s − 0.751·19-s + 1.08·20-s − 0.995·21-s − 2.41·22-s − 0.487·23-s − 0.164·24-s − 0.177·25-s − 2.11·26-s − 0.192·27-s + 2.05·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8007\)    =    \(3 \cdot 17 \cdot 157\)
Sign: $-1$
Analytic conductor: \(63.9362\)
Root analytic conductor: \(7.99601\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8007,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
17 \( 1 - T \)
157 \( 1 - T \)
good2 \( 1 - 2.09T + 2T^{2} \)
5 \( 1 - 2.02T + 5T^{2} \)
7 \( 1 - 4.56T + 7T^{2} \)
11 \( 1 + 5.41T + 11T^{2} \)
13 \( 1 + 5.15T + 13T^{2} \)
19 \( 1 + 3.27T + 19T^{2} \)
23 \( 1 + 2.33T + 23T^{2} \)
29 \( 1 - 3.91T + 29T^{2} \)
31 \( 1 + 2.52T + 31T^{2} \)
37 \( 1 + 9.27T + 37T^{2} \)
41 \( 1 + 6.58T + 41T^{2} \)
43 \( 1 + 12.3T + 43T^{2} \)
47 \( 1 + 5.98T + 47T^{2} \)
53 \( 1 - 4.67T + 53T^{2} \)
59 \( 1 - 8.19T + 59T^{2} \)
61 \( 1 - 0.225T + 61T^{2} \)
67 \( 1 - 14.7T + 67T^{2} \)
71 \( 1 + 9.65T + 71T^{2} \)
73 \( 1 + 5.16T + 73T^{2} \)
79 \( 1 - 3.36T + 79T^{2} \)
83 \( 1 + 3.93T + 83T^{2} \)
89 \( 1 + 8.53T + 89T^{2} \)
97 \( 1 - 1.86T + 97T^{2} \)
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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

−7.20730851371048017229619104306, −6.64189164461567315845538518124, −5.55126771490726818906568333486, −5.33524617546964898454619683272, −4.90845567348965206621069905659, −4.30729036723555035607823172224, −3.14042892102125572057598697980, −2.13840245069313023126041221937, −1.89005511664161400207968704028, 0, 1.89005511664161400207968704028, 2.13840245069313023126041221937, 3.14042892102125572057598697980, 4.30729036723555035607823172224, 4.90845567348965206621069905659, 5.33524617546964898454619683272, 5.55126771490726818906568333486, 6.64189164461567315845538518124, 7.20730851371048017229619104306

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