Properties

Label 2-4034-1.1-c1-0-84
Degree $2$
Conductor $4034$
Sign $-1$
Analytic cond. $32.2116$
Root an. cond. $5.67553$
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-s − 3.27·3-s + 4-s + 2.11·5-s + 3.27·6-s − 0.124·7-s − 8-s + 7.75·9-s − 2.11·10-s − 5.51·11-s − 3.27·12-s + 4.44·13-s + 0.124·14-s − 6.95·15-s + 16-s − 0.642·17-s − 7.75·18-s + 0.147·19-s + 2.11·20-s + 0.409·21-s + 5.51·22-s + 3.37·23-s + 3.27·24-s − 0.508·25-s − 4.44·26-s − 15.6·27-s − 0.124·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.89·3-s + 0.5·4-s + 0.947·5-s + 1.33·6-s − 0.0472·7-s − 0.353·8-s + 2.58·9-s − 0.670·10-s − 1.66·11-s − 0.946·12-s + 1.23·13-s + 0.0333·14-s − 1.79·15-s + 0.250·16-s − 0.155·17-s − 1.82·18-s + 0.0338·19-s + 0.473·20-s + 0.0894·21-s + 1.17·22-s + 0.704·23-s + 0.669·24-s − 0.101·25-s − 0.871·26-s − 3.00·27-s − 0.0236·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4034\)    =    \(2 \cdot 2017\)
Sign: $-1$
Analytic conductor: \(32.2116\)
Root analytic conductor: \(5.67553\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4034,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
2017 \( 1 + T \)
good3 \( 1 + 3.27T + 3T^{2} \)
5 \( 1 - 2.11T + 5T^{2} \)
7 \( 1 + 0.124T + 7T^{2} \)
11 \( 1 + 5.51T + 11T^{2} \)
13 \( 1 - 4.44T + 13T^{2} \)
17 \( 1 + 0.642T + 17T^{2} \)
19 \( 1 - 0.147T + 19T^{2} \)
23 \( 1 - 3.37T + 23T^{2} \)
29 \( 1 - 2.66T + 29T^{2} \)
31 \( 1 - 5.69T + 31T^{2} \)
37 \( 1 + 7.36T + 37T^{2} \)
41 \( 1 + 6.82T + 41T^{2} \)
43 \( 1 + 8.47T + 43T^{2} \)
47 \( 1 + 2.64T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 - 4.76T + 59T^{2} \)
61 \( 1 + 9.35T + 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 - 1.77T + 73T^{2} \)
79 \( 1 + 16.4T + 79T^{2} \)
83 \( 1 - 8.04T + 83T^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 + 5.37T + 97T^{2} \)
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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

−8.076180898826760411747338792397, −7.10672293233338863391879230740, −6.50616621284085703260976474846, −5.91780738164044301065638257858, −5.28432555120921881498894053682, −4.73464161865317399472050683863, −3.32359467798954020857131332253, −2.02950792366550579517653151253, −1.13532261385657021658854009011, 0, 1.13532261385657021658854009011, 2.02950792366550579517653151253, 3.32359467798954020857131332253, 4.73464161865317399472050683863, 5.28432555120921881498894053682, 5.91780738164044301065638257858, 6.50616621284085703260976474846, 7.10672293233338863391879230740, 8.076180898826760411747338792397

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