Properties

Label 2-2013-1.1-c3-0-204
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  + 2.20·2-s − 3·3-s − 3.15·4-s − 2.03·5-s − 6.60·6-s + 14.0·7-s − 24.5·8-s + 9·9-s − 4.48·10-s − 11·11-s + 9.46·12-s − 37.4·13-s + 30.9·14-s + 6.11·15-s − 28.7·16-s + 63.3·17-s + 19.8·18-s + 30.1·19-s + 6.43·20-s − 42.2·21-s − 24.2·22-s + 125.·23-s + 73.6·24-s − 120.·25-s − 82.4·26-s − 27·27-s − 44.4·28-s + ⋯
L(s)  = 1  + 0.778·2-s − 0.577·3-s − 0.394·4-s − 0.182·5-s − 0.449·6-s + 0.759·7-s − 1.08·8-s + 0.333·9-s − 0.141·10-s − 0.301·11-s + 0.227·12-s − 0.799·13-s + 0.591·14-s + 0.105·15-s − 0.449·16-s + 0.903·17-s + 0.259·18-s + 0.364·19-s + 0.0719·20-s − 0.438·21-s − 0.234·22-s + 1.13·23-s + 0.626·24-s − 0.966·25-s − 0.622·26-s − 0.192·27-s − 0.299·28-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 \( 1 - 2.20T + 8T^{2} \)
5 \( 1 + 2.03T + 125T^{2} \)
7 \( 1 - 14.0T + 343T^{2} \)
13 \( 1 + 37.4T + 2.19e3T^{2} \)
17 \( 1 - 63.3T + 4.91e3T^{2} \)
19 \( 1 - 30.1T + 6.85e3T^{2} \)
23 \( 1 - 125.T + 1.21e4T^{2} \)
29 \( 1 + 129.T + 2.43e4T^{2} \)
31 \( 1 - 179.T + 2.97e4T^{2} \)
37 \( 1 - 116.T + 5.06e4T^{2} \)
41 \( 1 + 238.T + 6.89e4T^{2} \)
43 \( 1 - 124.T + 7.95e4T^{2} \)
47 \( 1 - 324.T + 1.03e5T^{2} \)
53 \( 1 + 233.T + 1.48e5T^{2} \)
59 \( 1 + 378.T + 2.05e5T^{2} \)
67 \( 1 - 563.T + 3.00e5T^{2} \)
71 \( 1 - 591.T + 3.57e5T^{2} \)
73 \( 1 + 313.T + 3.89e5T^{2} \)
79 \( 1 - 714.T + 4.93e5T^{2} \)
83 \( 1 - 382.T + 5.71e5T^{2} \)
89 \( 1 + 536.T + 7.04e5T^{2} \)
97 \( 1 + 1.33e3T + 9.12e5T^{2} \)
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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

−8.215786171120421428199255598761, −7.65561142366790204375134431406, −6.68401239284097771665800019846, −5.64403302217164339038069764128, −5.16992485544712962718308053488, −4.50778279592444958958521718050, −3.60419674338252193050374337982, −2.57601819867533943656276394658, −1.15959844115238384235138795370, 0, 1.15959844115238384235138795370, 2.57601819867533943656276394658, 3.60419674338252193050374337982, 4.50778279592444958958521718050, 5.16992485544712962718308053488, 5.64403302217164339038069764128, 6.68401239284097771665800019846, 7.65561142366790204375134431406, 8.215786171120421428199255598761

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