Properties

Label 2-4029-1.1-c1-0-138
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  − 0.193·2-s − 3-s − 1.96·4-s + 0.930·5-s + 0.193·6-s + 3.24·7-s + 0.765·8-s + 9-s − 0.179·10-s − 0.492·11-s + 1.96·12-s + 0.350·13-s − 0.625·14-s − 0.930·15-s + 3.77·16-s − 17-s − 0.193·18-s + 0.0213·19-s − 1.82·20-s − 3.24·21-s + 0.0951·22-s − 7.67·23-s − 0.765·24-s − 4.13·25-s − 0.0676·26-s − 27-s − 6.35·28-s + ⋯
L(s)  = 1  − 0.136·2-s − 0.577·3-s − 0.981·4-s + 0.416·5-s + 0.0788·6-s + 1.22·7-s + 0.270·8-s + 0.333·9-s − 0.0568·10-s − 0.148·11-s + 0.566·12-s + 0.0971·13-s − 0.167·14-s − 0.240·15-s + 0.944·16-s − 0.242·17-s − 0.0455·18-s + 0.00490·19-s − 0.408·20-s − 0.707·21-s + 0.0202·22-s − 1.60·23-s − 0.156·24-s − 0.826·25-s − 0.0132·26-s − 0.192·27-s − 1.20·28-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 \( 1 + 0.193T + 2T^{2} \)
5 \( 1 - 0.930T + 5T^{2} \)
7 \( 1 - 3.24T + 7T^{2} \)
11 \( 1 + 0.492T + 11T^{2} \)
13 \( 1 - 0.350T + 13T^{2} \)
19 \( 1 - 0.0213T + 19T^{2} \)
23 \( 1 + 7.67T + 23T^{2} \)
29 \( 1 - 5.43T + 29T^{2} \)
31 \( 1 + 7.94T + 31T^{2} \)
37 \( 1 - 1.05T + 37T^{2} \)
41 \( 1 - 0.961T + 41T^{2} \)
43 \( 1 + 9.97T + 43T^{2} \)
47 \( 1 + 6.17T + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 - 7.91T + 59T^{2} \)
61 \( 1 - 0.289T + 61T^{2} \)
67 \( 1 - 14.7T + 67T^{2} \)
71 \( 1 + 1.17T + 71T^{2} \)
73 \( 1 + 0.851T + 73T^{2} \)
83 \( 1 + 3.79T + 83T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 + 2.22T + 97T^{2} \)
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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

−8.258023970907782660684165852310, −7.49820207861235429284374638753, −6.52192423404416746550728640940, −5.57496447338063675315765710193, −5.20813329600295929002010058831, −4.35106724532815930736876380265, −3.73334556514516051261091613233, −2.17360102342191557734944470630, −1.34224369825606489527226759971, 0, 1.34224369825606489527226759971, 2.17360102342191557734944470630, 3.73334556514516051261091613233, 4.35106724532815930736876380265, 5.20813329600295929002010058831, 5.57496447338063675315765710193, 6.52192423404416746550728640940, 7.49820207861235429284374638753, 8.258023970907782660684165852310

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