Properties

Label 2-4034-1.1-c1-0-10
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.09·3-s + 4-s − 0.358·5-s + 1.09·6-s − 2.93·7-s − 8-s − 1.79·9-s + 0.358·10-s − 2.70·11-s − 1.09·12-s + 0.734·13-s + 2.93·14-s + 0.393·15-s + 16-s + 0.422·17-s + 1.79·18-s + 8.46·19-s − 0.358·20-s + 3.21·21-s + 2.70·22-s − 8.04·23-s + 1.09·24-s − 4.87·25-s − 0.734·26-s + 5.25·27-s − 2.93·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.632·3-s + 0.5·4-s − 0.160·5-s + 0.447·6-s − 1.10·7-s − 0.353·8-s − 0.599·9-s + 0.113·10-s − 0.814·11-s − 0.316·12-s + 0.203·13-s + 0.783·14-s + 0.101·15-s + 0.250·16-s + 0.102·17-s + 0.423·18-s + 1.94·19-s − 0.0801·20-s + 0.700·21-s + 0.576·22-s − 1.67·23-s + 0.223·24-s − 0.974·25-s − 0.144·26-s + 1.01·27-s − 0.553·28-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: \(0\)
Selberg data: \((2,\ 4034,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2958611754\)
\(L(\frac12)\) \(\approx\) \(0.2958611754\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.09T + 3T^{2} \)
5 \( 1 + 0.358T + 5T^{2} \)
7 \( 1 + 2.93T + 7T^{2} \)
11 \( 1 + 2.70T + 11T^{2} \)
13 \( 1 - 0.734T + 13T^{2} \)
17 \( 1 - 0.422T + 17T^{2} \)
19 \( 1 - 8.46T + 19T^{2} \)
23 \( 1 + 8.04T + 23T^{2} \)
29 \( 1 + 6.64T + 29T^{2} \)
31 \( 1 - 3.26T + 31T^{2} \)
37 \( 1 + 3.52T + 37T^{2} \)
41 \( 1 + 4.23T + 41T^{2} \)
43 \( 1 + 0.269T + 43T^{2} \)
47 \( 1 + 13.2T + 47T^{2} \)
53 \( 1 - 2.29T + 53T^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 - 5.42T + 61T^{2} \)
67 \( 1 + 2.25T + 67T^{2} \)
71 \( 1 + 6.19T + 71T^{2} \)
73 \( 1 + 14.6T + 73T^{2} \)
79 \( 1 - 7.19T + 79T^{2} \)
83 \( 1 + 2.45T + 83T^{2} \)
89 \( 1 - 5.76T + 89T^{2} \)
97 \( 1 + 7.90T + 97T^{2} \)
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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

−8.360133825253820722929920520985, −7.74619889637818839991150636622, −7.08472487337152806567955169536, −6.05326993324415750079860596583, −5.82560087117676928973659928562, −4.89083345081967114247851975953, −3.51931032813186989225899233048, −3.03550154057286062685792888880, −1.80272101936683949071532318888, −0.34128000299908444164992810245, 0.34128000299908444164992810245, 1.80272101936683949071532318888, 3.03550154057286062685792888880, 3.51931032813186989225899233048, 4.89083345081967114247851975953, 5.82560087117676928973659928562, 6.05326993324415750079860596583, 7.08472487337152806567955169536, 7.74619889637818839991150636622, 8.360133825253820722929920520985

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