Properties

Label 2-1008-1.1-c5-0-50
Degree $2$
Conductor $1008$
Sign $-1$
Analytic cond. $161.666$
Root an. cond. $12.7148$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s − 49·7-s − 108·11-s − 346·13-s + 1.39e3·17-s + 1.01e3·19-s − 1.53e3·23-s − 3.08e3·25-s + 3.76e3·29-s + 736·31-s + 294·35-s + 2.05e3·37-s + 1.55e4·41-s − 1.10e4·43-s + 4.56e3·47-s + 2.40e3·49-s + 7.96e3·53-s + 648·55-s − 7.02e3·59-s + 2.68e4·61-s + 2.07e3·65-s − 5.21e4·67-s − 2.54e3·71-s − 9.76e3·73-s + 5.29e3·77-s − 6.86e4·79-s − 6.16e4·83-s + ⋯
L(s)  = 1  − 0.107·5-s − 0.377·7-s − 0.269·11-s − 0.567·13-s + 1.17·17-s + 0.643·19-s − 0.605·23-s − 0.988·25-s + 0.830·29-s + 0.137·31-s + 0.0405·35-s + 0.246·37-s + 1.44·41-s − 0.910·43-s + 0.301·47-s + 1/7·49-s + 0.389·53-s + 0.0288·55-s − 0.262·59-s + 0.924·61-s + 0.0609·65-s − 1.41·67-s − 0.0598·71-s − 0.214·73-s + 0.101·77-s − 1.23·79-s − 0.982·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(161.666\)
Root analytic conductor: \(12.7148\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1008,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + p^{2} T \)
good5 \( 1 + 6 T + p^{5} T^{2} \)
11 \( 1 + 108 T + p^{5} T^{2} \)
13 \( 1 + 346 T + p^{5} T^{2} \)
17 \( 1 - 1398 T + p^{5} T^{2} \)
19 \( 1 - 1012 T + p^{5} T^{2} \)
23 \( 1 + 1536 T + p^{5} T^{2} \)
29 \( 1 - 3762 T + p^{5} T^{2} \)
31 \( 1 - 736 T + p^{5} T^{2} \)
37 \( 1 - 2054 T + p^{5} T^{2} \)
41 \( 1 - 15534 T + p^{5} T^{2} \)
43 \( 1 + 11036 T + p^{5} T^{2} \)
47 \( 1 - 4560 T + p^{5} T^{2} \)
53 \( 1 - 7962 T + p^{5} T^{2} \)
59 \( 1 + 7020 T + p^{5} T^{2} \)
61 \( 1 - 26870 T + p^{5} T^{2} \)
67 \( 1 + 52148 T + p^{5} T^{2} \)
71 \( 1 + 2544 T + p^{5} T^{2} \)
73 \( 1 + 9766 T + p^{5} T^{2} \)
79 \( 1 + 68672 T + p^{5} T^{2} \)
83 \( 1 + 61668 T + p^{5} T^{2} \)
89 \( 1 - 41454 T + p^{5} T^{2} \)
97 \( 1 + 111262 T + p^{5} T^{2} \)
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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

−8.826787580058151594080234015351, −7.84226691476574103383633820443, −7.31598798276807002331315817806, −6.15951019485288211337274638472, −5.44745556836832910831261398068, −4.39106776059521212161625914786, −3.37412474085905702284477905087, −2.45656541608712261248319071056, −1.14881882715057224941762489990, 0, 1.14881882715057224941762489990, 2.45656541608712261248319071056, 3.37412474085905702284477905087, 4.39106776059521212161625914786, 5.44745556836832910831261398068, 6.15951019485288211337274638472, 7.31598798276807002331315817806, 7.84226691476574103383633820443, 8.826787580058151594080234015351

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