Properties

Label 2-4034-1.1-c1-0-94
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 − 1.52·3-s + 4-s + 3.51·5-s + 1.52·6-s − 1.62·7-s − 8-s − 0.661·9-s − 3.51·10-s − 5.23·11-s − 1.52·12-s − 0.449·13-s + 1.62·14-s − 5.37·15-s + 16-s + 1.78·17-s + 0.661·18-s − 1.93·19-s + 3.51·20-s + 2.47·21-s + 5.23·22-s − 0.570·23-s + 1.52·24-s + 7.36·25-s + 0.449·26-s + 5.59·27-s − 1.62·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.882·3-s + 0.5·4-s + 1.57·5-s + 0.624·6-s − 0.612·7-s − 0.353·8-s − 0.220·9-s − 1.11·10-s − 1.57·11-s − 0.441·12-s − 0.124·13-s + 0.433·14-s − 1.38·15-s + 0.250·16-s + 0.433·17-s + 0.155·18-s − 0.444·19-s + 0.786·20-s + 0.541·21-s + 1.11·22-s − 0.119·23-s + 0.312·24-s + 1.47·25-s + 0.0881·26-s + 1.07·27-s − 0.306·28-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 \( 1 + 1.52T + 3T^{2} \)
5 \( 1 - 3.51T + 5T^{2} \)
7 \( 1 + 1.62T + 7T^{2} \)
11 \( 1 + 5.23T + 11T^{2} \)
13 \( 1 + 0.449T + 13T^{2} \)
17 \( 1 - 1.78T + 17T^{2} \)
19 \( 1 + 1.93T + 19T^{2} \)
23 \( 1 + 0.570T + 23T^{2} \)
29 \( 1 - 2.33T + 29T^{2} \)
31 \( 1 - 7.15T + 31T^{2} \)
37 \( 1 - 0.308T + 37T^{2} \)
41 \( 1 - 12.6T + 41T^{2} \)
43 \( 1 - 8.05T + 43T^{2} \)
47 \( 1 + 1.84T + 47T^{2} \)
53 \( 1 + 1.51T + 53T^{2} \)
59 \( 1 - 0.0896T + 59T^{2} \)
61 \( 1 + 8.72T + 61T^{2} \)
67 \( 1 + 2.67T + 67T^{2} \)
71 \( 1 + 6.78T + 71T^{2} \)
73 \( 1 + 8.78T + 73T^{2} \)
79 \( 1 + 6.71T + 79T^{2} \)
83 \( 1 + 4.78T + 83T^{2} \)
89 \( 1 - 2.06T + 89T^{2} \)
97 \( 1 - 0.506T + 97T^{2} \)
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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

−8.118472245111541666372372198769, −7.32672078058449420385204600173, −6.31193875797395342423555677442, −6.00987601061867723168884734217, −5.41527662612989068552403810800, −4.58048170708254752862762891509, −2.81646307459244043940270841698, −2.55585481000470181138821045455, −1.20900834029276177869573199913, 0, 1.20900834029276177869573199913, 2.55585481000470181138821045455, 2.81646307459244043940270841698, 4.58048170708254752862762891509, 5.41527662612989068552403810800, 6.00987601061867723168884734217, 6.31193875797395342423555677442, 7.32672078058449420385204600173, 8.118472245111541666372372198769

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