Properties

Label 2-1008-1.1-c3-0-30
Degree $2$
Conductor $1008$
Sign $-1$
Analytic cond. $59.4739$
Root an. cond. $7.71193$
Motivic weight $3$
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 − 7·7-s + 30·11-s + 2·13-s − 66·17-s + 52·19-s + 114·23-s − 89·25-s − 72·29-s + 196·31-s + 42·35-s − 286·37-s + 378·41-s − 164·43-s − 228·47-s + 49·49-s + 348·53-s − 180·55-s − 348·59-s − 106·61-s − 12·65-s − 596·67-s + 630·71-s − 1.04e3·73-s − 210·77-s + 88·79-s − 1.44e3·83-s + ⋯
L(s)  = 1  − 0.536·5-s − 0.377·7-s + 0.822·11-s + 0.0426·13-s − 0.941·17-s + 0.627·19-s + 1.03·23-s − 0.711·25-s − 0.461·29-s + 1.13·31-s + 0.202·35-s − 1.27·37-s + 1.43·41-s − 0.581·43-s − 0.707·47-s + 1/7·49-s + 0.901·53-s − 0.441·55-s − 0.767·59-s − 0.222·61-s − 0.0228·65-s − 1.08·67-s + 1.05·71-s − 1.67·73-s − 0.310·77-s + 0.125·79-s − 1.90·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(59.4739\)
Root analytic conductor: \(7.71193\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1008} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1008,\ (\ :3/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + p T \)
good5 \( 1 + 6 T + p^{3} T^{2} \)
11 \( 1 - 30 T + p^{3} T^{2} \)
13 \( 1 - 2 T + p^{3} T^{2} \)
17 \( 1 + 66 T + p^{3} T^{2} \)
19 \( 1 - 52 T + p^{3} T^{2} \)
23 \( 1 - 114 T + p^{3} T^{2} \)
29 \( 1 + 72 T + p^{3} T^{2} \)
31 \( 1 - 196 T + p^{3} T^{2} \)
37 \( 1 + 286 T + p^{3} T^{2} \)
41 \( 1 - 378 T + p^{3} T^{2} \)
43 \( 1 + 164 T + p^{3} T^{2} \)
47 \( 1 + 228 T + p^{3} T^{2} \)
53 \( 1 - 348 T + p^{3} T^{2} \)
59 \( 1 + 348 T + p^{3} T^{2} \)
61 \( 1 + 106 T + p^{3} T^{2} \)
67 \( 1 + 596 T + p^{3} T^{2} \)
71 \( 1 - 630 T + p^{3} T^{2} \)
73 \( 1 + 1042 T + p^{3} T^{2} \)
79 \( 1 - 88 T + p^{3} T^{2} \)
83 \( 1 + 1440 T + p^{3} T^{2} \)
89 \( 1 + 1374 T + p^{3} T^{2} \)
97 \( 1 + 34 T + p^{3} T^{2} \)
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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

−9.142307530430375087112188748552, −8.435247862143007885165249499112, −7.37888557783176199383496876521, −6.72930965006540657643165739014, −5.78735511331249513168835290613, −4.63125802336541800294473253355, −3.79994943274037528780189034050, −2.79544116992844498314145430436, −1.35715844982625394239371650552, 0, 1.35715844982625394239371650552, 2.79544116992844498314145430436, 3.79994943274037528780189034050, 4.63125802336541800294473253355, 5.78735511331249513168835290613, 6.72930965006540657643165739014, 7.37888557783176199383496876521, 8.435247862143007885165249499112, 9.142307530430375087112188748552

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