Properties

Label 2-8007-1.1-c1-0-291
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  − 0.679·2-s − 3-s − 1.53·4-s + 2.83·5-s + 0.679·6-s − 0.718·7-s + 2.40·8-s + 9-s − 1.92·10-s − 0.429·11-s + 1.53·12-s + 0.378·13-s + 0.488·14-s − 2.83·15-s + 1.44·16-s + 17-s − 0.679·18-s − 0.638·19-s − 4.35·20-s + 0.718·21-s + 0.291·22-s + 4.51·23-s − 2.40·24-s + 3.01·25-s − 0.257·26-s − 27-s + 1.10·28-s + ⋯
L(s)  = 1  − 0.480·2-s − 0.577·3-s − 0.768·4-s + 1.26·5-s + 0.277·6-s − 0.271·7-s + 0.850·8-s + 0.333·9-s − 0.608·10-s − 0.129·11-s + 0.443·12-s + 0.105·13-s + 0.130·14-s − 0.730·15-s + 0.360·16-s + 0.242·17-s − 0.160·18-s − 0.146·19-s − 0.973·20-s + 0.156·21-s + 0.0622·22-s + 0.942·23-s − 0.490·24-s + 0.602·25-s − 0.0505·26-s − 0.192·27-s + 0.208·28-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 \( 1 + 0.679T + 2T^{2} \)
5 \( 1 - 2.83T + 5T^{2} \)
7 \( 1 + 0.718T + 7T^{2} \)
11 \( 1 + 0.429T + 11T^{2} \)
13 \( 1 - 0.378T + 13T^{2} \)
19 \( 1 + 0.638T + 19T^{2} \)
23 \( 1 - 4.51T + 23T^{2} \)
29 \( 1 - 0.186T + 29T^{2} \)
31 \( 1 + 6.25T + 31T^{2} \)
37 \( 1 + 2.51T + 37T^{2} \)
41 \( 1 + 3.44T + 41T^{2} \)
43 \( 1 - 8.07T + 43T^{2} \)
47 \( 1 + 7.10T + 47T^{2} \)
53 \( 1 + 6.64T + 53T^{2} \)
59 \( 1 + 2.16T + 59T^{2} \)
61 \( 1 - 1.34T + 61T^{2} \)
67 \( 1 + 0.865T + 67T^{2} \)
71 \( 1 - 8.28T + 71T^{2} \)
73 \( 1 + 14.5T + 73T^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 + 8.22T + 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 - 4.45T + 97T^{2} \)
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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

−7.52096000031611252937930819182, −6.71381562593564881850865183070, −6.08620490210244481634688069290, −5.31478638355218653184183500317, −4.97060401460764871075871870894, −3.98381800729208326427345626941, −3.07740077459449010290436214282, −1.91540379519828004765634187019, −1.18730710758410228851540384567, 0, 1.18730710758410228851540384567, 1.91540379519828004765634187019, 3.07740077459449010290436214282, 3.98381800729208326427345626941, 4.97060401460764871075871870894, 5.31478638355218653184183500317, 6.08620490210244481634688069290, 6.71381562593564881850865183070, 7.52096000031611252937930819182

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