Properties

Label 2-4034-1.1-c1-0-35
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 − 1.90·3-s + 4-s − 3.49·5-s + 1.90·6-s − 2.88·7-s − 8-s + 0.629·9-s + 3.49·10-s − 2.01·11-s − 1.90·12-s − 5.40·13-s + 2.88·14-s + 6.66·15-s + 16-s + 4.25·17-s − 0.629·18-s + 2.92·19-s − 3.49·20-s + 5.49·21-s + 2.01·22-s − 2.94·23-s + 1.90·24-s + 7.24·25-s + 5.40·26-s + 4.51·27-s − 2.88·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.09·3-s + 0.5·4-s − 1.56·5-s + 0.777·6-s − 1.09·7-s − 0.353·8-s + 0.209·9-s + 1.10·10-s − 0.606·11-s − 0.549·12-s − 1.50·13-s + 0.770·14-s + 1.72·15-s + 0.250·16-s + 1.03·17-s − 0.148·18-s + 0.670·19-s − 0.782·20-s + 1.19·21-s + 0.428·22-s − 0.615·23-s + 0.388·24-s + 1.44·25-s + 1.06·26-s + 0.869·27-s − 0.545·28-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 \( 1 + 1.90T + 3T^{2} \)
5 \( 1 + 3.49T + 5T^{2} \)
7 \( 1 + 2.88T + 7T^{2} \)
11 \( 1 + 2.01T + 11T^{2} \)
13 \( 1 + 5.40T + 13T^{2} \)
17 \( 1 - 4.25T + 17T^{2} \)
19 \( 1 - 2.92T + 19T^{2} \)
23 \( 1 + 2.94T + 23T^{2} \)
29 \( 1 + 3.50T + 29T^{2} \)
31 \( 1 + 7.26T + 31T^{2} \)
37 \( 1 - 7.06T + 37T^{2} \)
41 \( 1 + 2.34T + 41T^{2} \)
43 \( 1 - 4.99T + 43T^{2} \)
47 \( 1 - 8.87T + 47T^{2} \)
53 \( 1 - 7.95T + 53T^{2} \)
59 \( 1 - 11.9T + 59T^{2} \)
61 \( 1 + 0.563T + 61T^{2} \)
67 \( 1 + 0.351T + 67T^{2} \)
71 \( 1 + 4.94T + 71T^{2} \)
73 \( 1 + 1.76T + 73T^{2} \)
79 \( 1 + 4.78T + 79T^{2} \)
83 \( 1 + 7.48T + 83T^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 - 0.00634T + 97T^{2} \)
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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

−7.77492706371925939263596550210, −7.45621724354734482653715868720, −6.87042966289074055217965076774, −5.78604306180625219091846176812, −5.32838243621026401103625812670, −4.23510327956857365599242514548, −3.36748189747382611157344955604, −2.54107702928981780944954387491, −0.69967277835317185102181169026, 0, 0.69967277835317185102181169026, 2.54107702928981780944954387491, 3.36748189747382611157344955604, 4.23510327956857365599242514548, 5.32838243621026401103625812670, 5.78604306180625219091846176812, 6.87042966289074055217965076774, 7.45621724354734482653715868720, 7.77492706371925939263596550210

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