Properties

Label 2-4030-1.1-c1-0-93
Degree $2$
Conductor $4030$
Sign $-1$
Analytic cond. $32.1797$
Root an. cond. $5.67271$
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 − 2.06·3-s + 4-s + 5-s − 2.06·6-s + 0.701·7-s + 8-s + 1.27·9-s + 10-s − 4.56·11-s − 2.06·12-s − 13-s + 0.701·14-s − 2.06·15-s + 16-s + 7.12·17-s + 1.27·18-s − 6.61·19-s + 20-s − 1.45·21-s − 4.56·22-s + 3.53·23-s − 2.06·24-s + 25-s − 26-s + 3.55·27-s + 0.701·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.19·3-s + 0.5·4-s + 0.447·5-s − 0.844·6-s + 0.265·7-s + 0.353·8-s + 0.426·9-s + 0.316·10-s − 1.37·11-s − 0.597·12-s − 0.277·13-s + 0.187·14-s − 0.534·15-s + 0.250·16-s + 1.72·17-s + 0.301·18-s − 1.51·19-s + 0.223·20-s − 0.316·21-s − 0.974·22-s + 0.737·23-s − 0.422·24-s + 0.200·25-s − 0.196·26-s + 0.685·27-s + 0.132·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4030\)    =    \(2 \cdot 5 \cdot 13 \cdot 31\)
Sign: $-1$
Analytic conductor: \(32.1797\)
Root analytic conductor: \(5.67271\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4030,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 + T \)
31 \( 1 + T \)
good3 \( 1 + 2.06T + 3T^{2} \)
7 \( 1 - 0.701T + 7T^{2} \)
11 \( 1 + 4.56T + 11T^{2} \)
17 \( 1 - 7.12T + 17T^{2} \)
19 \( 1 + 6.61T + 19T^{2} \)
23 \( 1 - 3.53T + 23T^{2} \)
29 \( 1 + 1.57T + 29T^{2} \)
37 \( 1 + 1.23T + 37T^{2} \)
41 \( 1 - 3.18T + 41T^{2} \)
43 \( 1 + 5.91T + 43T^{2} \)
47 \( 1 + 5.32T + 47T^{2} \)
53 \( 1 + 3.23T + 53T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 - 2.53T + 61T^{2} \)
67 \( 1 - 5.14T + 67T^{2} \)
71 \( 1 + 3.62T + 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 + 1.62T + 79T^{2} \)
83 \( 1 + 6.10T + 83T^{2} \)
89 \( 1 - 13.7T + 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−7.88327007713128977658157475422, −7.19297401196295751549720939465, −6.24763426462944770271954027705, −5.81383049798487493026225372432, −5.02131268623076049431122666088, −4.76335819656839770594136562883, −3.41395779453875954296860723131, −2.56666051846758708496482413492, −1.44096026793157959599940722278, 0, 1.44096026793157959599940722278, 2.56666051846758708496482413492, 3.41395779453875954296860723131, 4.76335819656839770594136562883, 5.02131268623076049431122666088, 5.81383049798487493026225372432, 6.24763426462944770271954027705, 7.19297401196295751549720939465, 7.88327007713128977658157475422

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