Properties

Label 2-4029-1.1-c1-0-119
Degree $2$
Conductor $4029$
Sign $-1$
Analytic cond. $32.1717$
Root an. cond. $5.67201$
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  − 2.72·2-s + 3-s + 5.40·4-s − 2.75·5-s − 2.72·6-s + 0.631·7-s − 9.27·8-s + 9-s + 7.51·10-s + 0.617·11-s + 5.40·12-s + 2.46·13-s − 1.72·14-s − 2.75·15-s + 14.4·16-s − 17-s − 2.72·18-s + 2.03·19-s − 14.9·20-s + 0.631·21-s − 1.68·22-s + 3.83·23-s − 9.27·24-s + 2.61·25-s − 6.70·26-s + 27-s + 3.41·28-s + ⋯
L(s)  = 1  − 1.92·2-s + 0.577·3-s + 2.70·4-s − 1.23·5-s − 1.11·6-s + 0.238·7-s − 3.27·8-s + 0.333·9-s + 2.37·10-s + 0.186·11-s + 1.56·12-s + 0.683·13-s − 0.459·14-s − 0.712·15-s + 3.60·16-s − 0.242·17-s − 0.641·18-s + 0.466·19-s − 3.33·20-s + 0.137·21-s − 0.358·22-s + 0.800·23-s − 1.89·24-s + 0.523·25-s − 1.31·26-s + 0.192·27-s + 0.645·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $-1$
Analytic conductor: \(32.1717\)
Root analytic conductor: \(5.67201\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4029,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
79 \( 1 - T \)
good2 \( 1 + 2.72T + 2T^{2} \)
5 \( 1 + 2.75T + 5T^{2} \)
7 \( 1 - 0.631T + 7T^{2} \)
11 \( 1 - 0.617T + 11T^{2} \)
13 \( 1 - 2.46T + 13T^{2} \)
19 \( 1 - 2.03T + 19T^{2} \)
23 \( 1 - 3.83T + 23T^{2} \)
29 \( 1 + 8.75T + 29T^{2} \)
31 \( 1 + 3.85T + 31T^{2} \)
37 \( 1 + 1.46T + 37T^{2} \)
41 \( 1 + 4.27T + 41T^{2} \)
43 \( 1 + 5.41T + 43T^{2} \)
47 \( 1 - 6.44T + 47T^{2} \)
53 \( 1 + 6.48T + 53T^{2} \)
59 \( 1 - 5.45T + 59T^{2} \)
61 \( 1 - 1.97T + 61T^{2} \)
67 \( 1 + 5.16T + 67T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + 1.27T + 73T^{2} \)
83 \( 1 - 6.14T + 83T^{2} \)
89 \( 1 - 7.73T + 89T^{2} \)
97 \( 1 - 6.89T + 97T^{2} \)
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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

−8.088389257059713257393478327549, −7.66535638517349313081182535435, −7.08396587857958714668169780692, −6.36400015093689458044638068308, −5.19967790644752519191910586998, −3.78958253695917404423430631996, −3.27258831730530033545807698198, −2.10502500111265867576986125240, −1.20340289813640878301603129458, 0, 1.20340289813640878301603129458, 2.10502500111265867576986125240, 3.27258831730530033545807698198, 3.78958253695917404423430631996, 5.19967790644752519191910586998, 6.36400015093689458044638068308, 7.08396587857958714668169780692, 7.66535638517349313081182535435, 8.088389257059713257393478327549

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