Properties

Label 2-2013-1.1-c3-0-188
Degree $2$
Conductor $2013$
Sign $-1$
Analytic cond. $118.770$
Root an. cond. $10.8982$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.75·2-s − 3·3-s − 0.392·4-s + 6.83·5-s + 8.27·6-s − 1.91·7-s + 23.1·8-s + 9·9-s − 18.8·10-s + 11·11-s + 1.17·12-s + 3.15·13-s + 5.28·14-s − 20.5·15-s − 60.7·16-s + 79.2·17-s − 24.8·18-s + 27.1·19-s − 2.68·20-s + 5.74·21-s − 30.3·22-s − 127.·23-s − 69.4·24-s − 78.2·25-s − 8.71·26-s − 27·27-s + 0.751·28-s + ⋯
L(s)  = 1  − 0.975·2-s − 0.577·3-s − 0.0490·4-s + 0.611·5-s + 0.563·6-s − 0.103·7-s + 1.02·8-s + 0.333·9-s − 0.596·10-s + 0.301·11-s + 0.0283·12-s + 0.0673·13-s + 0.100·14-s − 0.353·15-s − 0.948·16-s + 1.13·17-s − 0.325·18-s + 0.328·19-s − 0.0300·20-s + 0.0597·21-s − 0.294·22-s − 1.15·23-s − 0.590·24-s − 0.625·25-s − 0.0657·26-s − 0.192·27-s + 0.00507·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $-1$
Analytic conductor: \(118.770\)
Root analytic conductor: \(10.8982\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2013,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
11 \( 1 - 11T \)
61 \( 1 + 61T \)
good2 \( 1 + 2.75T + 8T^{2} \)
5 \( 1 - 6.83T + 125T^{2} \)
7 \( 1 + 1.91T + 343T^{2} \)
13 \( 1 - 3.15T + 2.19e3T^{2} \)
17 \( 1 - 79.2T + 4.91e3T^{2} \)
19 \( 1 - 27.1T + 6.85e3T^{2} \)
23 \( 1 + 127.T + 1.21e4T^{2} \)
29 \( 1 - 6.05T + 2.43e4T^{2} \)
31 \( 1 - 311.T + 2.97e4T^{2} \)
37 \( 1 - 108.T + 5.06e4T^{2} \)
41 \( 1 + 252.T + 6.89e4T^{2} \)
43 \( 1 + 250.T + 7.95e4T^{2} \)
47 \( 1 + 482.T + 1.03e5T^{2} \)
53 \( 1 + 502.T + 1.48e5T^{2} \)
59 \( 1 - 655.T + 2.05e5T^{2} \)
67 \( 1 + 6.83T + 3.00e5T^{2} \)
71 \( 1 - 2.88T + 3.57e5T^{2} \)
73 \( 1 + 0.776T + 3.89e5T^{2} \)
79 \( 1 - 339.T + 4.93e5T^{2} \)
83 \( 1 - 515.T + 5.71e5T^{2} \)
89 \( 1 + 1.41e3T + 7.04e5T^{2} \)
97 \( 1 + 847.T + 9.12e5T^{2} \)
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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

−8.252373226639622975933004678333, −7.968423271465789582478815005647, −6.83199011420189276453540016489, −6.14460744674910779808540901791, −5.26096669248183597170583414716, −4.45026386745087441872822611997, −3.34597491149913814804640430095, −1.88962920914083750006301292527, −1.10091467545392046653949662084, 0, 1.10091467545392046653949662084, 1.88962920914083750006301292527, 3.34597491149913814804640430095, 4.45026386745087441872822611997, 5.26096669248183597170583414716, 6.14460744674910779808540901791, 6.83199011420189276453540016489, 7.968423271465789582478815005647, 8.252373226639622975933004678333

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