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Properties

Label 2-1008-252.103-c1-0-40
Degree $2$
Conductor $1008$
Sign $0.594 + 0.804i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.43 − 0.967i)3-s + (1.73 − 0.999i)5-s + (2.38 − 1.15i)7-s + (1.12 − 2.77i)9-s + (1.17 + 0.681i)11-s + (1.71 + 0.987i)13-s + (1.52 − 3.11i)15-s + (0.868 − 0.501i)17-s + (−0.774 + 1.34i)19-s + (2.30 − 3.96i)21-s + (−7.59 + 4.38i)23-s + (−0.500 + 0.867i)25-s + (−1.06 − 5.08i)27-s + (0.854 + 1.48i)29-s − 0.522·31-s + ⋯
L(s)  = 1  + (0.829 − 0.558i)3-s + (0.774 − 0.447i)5-s + (0.899 − 0.436i)7-s + (0.376 − 0.926i)9-s + (0.355 + 0.205i)11-s + (0.474 + 0.273i)13-s + (0.392 − 0.803i)15-s + (0.210 − 0.121i)17-s + (−0.177 + 0.307i)19-s + (0.503 − 0.864i)21-s + (−1.58 + 0.914i)23-s + (−0.100 + 0.173i)25-s + (−0.204 − 0.978i)27-s + (0.158 + 0.274i)29-s − 0.0937·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.594 + 0.804i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.594 + 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.714597773\)
\(L(\frac12)\) \(\approx\) \(2.714597773\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.43 + 0.967i)T \)
7 \( 1 + (-2.38 + 1.15i)T \)
good5 \( 1 + (-1.73 + 0.999i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.17 - 0.681i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.71 - 0.987i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.868 + 0.501i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.774 - 1.34i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.59 - 4.38i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.854 - 1.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.522T + 31T^{2} \)
37 \( 1 + (-1.53 + 2.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.386 + 0.223i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.49 - 3.17i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.63T + 47T^{2} \)
53 \( 1 + (-1.24 - 2.15i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 - 10.5iT - 61T^{2} \)
67 \( 1 + 6.50iT - 67T^{2} \)
71 \( 1 + 1.57iT - 71T^{2} \)
73 \( 1 + (-13.6 + 7.87i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 5.95iT - 79T^{2} \)
83 \( 1 + (4.80 + 8.32i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.10 + 0.640i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.31 + 4.80i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.630425258545871410961741719546, −9.030488570074251628237171550074, −8.103924420864418150122754384464, −7.55888572759370184833240709144, −6.48219642686266844281962624434, −5.62880960409557962275892650411, −4.43559569512828839768692072775, −3.51869161388241561887128124780, −1.98634417504864592116061484552, −1.37998169846960832183159775894, 1.78218633823306606201125412980, 2.61037732727475572981413999912, 3.80412858652389138156362584823, 4.76496200500679415149485997946, 5.77733605595675279583289799714, 6.62067903635117421612471614042, 7.983248716214442210945733599200, 8.352393157434067898530524760364, 9.285873268849209238564836989675, 10.07011649730642338951922513505

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