Properties

Label 2-1008-1.1-c5-0-26
Degree $2$
Conductor $1008$
Sign $1$
Analytic cond. $161.666$
Root an. cond. $12.7148$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 84.8·5-s + 49·7-s − 634.·11-s + 895.·13-s − 2.05e3·17-s − 2.45e3·19-s − 569.·23-s + 4.07e3·25-s − 1.47e3·29-s + 2.00e3·31-s + 4.15e3·35-s + 4.86e3·37-s + 1.72e4·41-s + 1.54e4·43-s + 5.00e3·47-s + 2.40e3·49-s + 1.95e4·53-s − 5.37e4·55-s + 1.45e4·59-s + 3.57e3·61-s + 7.59e4·65-s + 4.14e4·67-s + 9.24e3·71-s − 4.13e4·73-s − 3.10e4·77-s + 3.79e4·79-s + 7.92e4·83-s + ⋯
L(s)  = 1  + 1.51·5-s + 0.377·7-s − 1.58·11-s + 1.46·13-s − 1.72·17-s − 1.55·19-s − 0.224·23-s + 1.30·25-s − 0.324·29-s + 0.375·31-s + 0.573·35-s + 0.583·37-s + 1.60·41-s + 1.27·43-s + 0.330·47-s + 0.142·49-s + 0.956·53-s − 2.39·55-s + 0.542·59-s + 0.122·61-s + 2.22·65-s + 1.12·67-s + 0.217·71-s − 0.908·73-s − 0.597·77-s + 0.684·79-s + 1.26·83-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.900452328\)
\(L(\frac12)\) \(\approx\) \(2.900452328\)
\(L(\frac{7}{2})\) 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 \( 1 - 49T \)
good5 \( 1 - 84.8T + 3.12e3T^{2} \)
11 \( 1 + 634.T + 1.61e5T^{2} \)
13 \( 1 - 895.T + 3.71e5T^{2} \)
17 \( 1 + 2.05e3T + 1.41e6T^{2} \)
19 \( 1 + 2.45e3T + 2.47e6T^{2} \)
23 \( 1 + 569.T + 6.43e6T^{2} \)
29 \( 1 + 1.47e3T + 2.05e7T^{2} \)
31 \( 1 - 2.00e3T + 2.86e7T^{2} \)
37 \( 1 - 4.86e3T + 6.93e7T^{2} \)
41 \( 1 - 1.72e4T + 1.15e8T^{2} \)
43 \( 1 - 1.54e4T + 1.47e8T^{2} \)
47 \( 1 - 5.00e3T + 2.29e8T^{2} \)
53 \( 1 - 1.95e4T + 4.18e8T^{2} \)
59 \( 1 - 1.45e4T + 7.14e8T^{2} \)
61 \( 1 - 3.57e3T + 8.44e8T^{2} \)
67 \( 1 - 4.14e4T + 1.35e9T^{2} \)
71 \( 1 - 9.24e3T + 1.80e9T^{2} \)
73 \( 1 + 4.13e4T + 2.07e9T^{2} \)
79 \( 1 - 3.79e4T + 3.07e9T^{2} \)
83 \( 1 - 7.92e4T + 3.93e9T^{2} \)
89 \( 1 - 9.25e4T + 5.58e9T^{2} \)
97 \( 1 - 1.75e5T + 8.58e9T^{2} \)
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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

−9.069690076093353070721977368096, −8.626726973657812063234531526907, −7.62680248551508267808985383849, −6.33713624594074903365243265872, −6.00069052712508487249987156802, −4.99500191226428086290024183404, −4.05537909450542803980544985706, −2.40506166684107135913916158445, −2.13403958853659367740588975696, −0.72408805334127802649174735598, 0.72408805334127802649174735598, 2.13403958853659367740588975696, 2.40506166684107135913916158445, 4.05537909450542803980544985706, 4.99500191226428086290024183404, 6.00069052712508487249987156802, 6.33713624594074903365243265872, 7.62680248551508267808985383849, 8.626726973657812063234531526907, 9.069690076093353070721977368096

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