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Properties

Label 2-1008-252.11-c1-0-39
Degree $2$
Conductor $1008$
Sign $-0.988 - 0.152i$
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  + (−1.67 + 0.422i)3-s + (−1.72 − 0.996i)5-s + (1.05 − 2.42i)7-s + (2.64 − 1.42i)9-s + (−1.23 − 2.14i)11-s + (−1.09 − 1.89i)13-s + (3.32 + 0.944i)15-s + (5.78 + 3.34i)17-s + (−6.14 + 3.54i)19-s + (−0.737 + 4.52i)21-s + (−2.03 + 3.53i)23-s + (−0.512 − 0.888i)25-s + (−3.83 + 3.50i)27-s + (7.02 + 4.05i)29-s − 5.85i·31-s + ⋯
L(s)  = 1  + (−0.969 + 0.244i)3-s + (−0.772 − 0.445i)5-s + (0.396 − 0.917i)7-s + (0.880 − 0.473i)9-s + (−0.373 − 0.647i)11-s + (−0.303 − 0.526i)13-s + (0.857 + 0.243i)15-s + (1.40 + 0.810i)17-s + (−1.41 + 0.814i)19-s + (−0.160 + 0.986i)21-s + (−0.425 + 0.736i)23-s + (−0.102 − 0.177i)25-s + (−0.738 + 0.674i)27-s + (1.30 + 0.753i)29-s − 1.05i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.988 - 0.152i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.988 - 0.152i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1744017791\)
\(L(\frac12)\) \(\approx\) \(0.1744017791\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 - 0.422i)T \)
7 \( 1 + (-1.05 + 2.42i)T \)
good5 \( 1 + (1.72 + 0.996i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.23 + 2.14i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.09 + 1.89i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-5.78 - 3.34i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.14 - 3.54i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.03 - 3.53i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.02 - 4.05i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.85iT - 31T^{2} \)
37 \( 1 + (4.48 + 7.77i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.86 - 3.38i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.34 + 2.51i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 13.2T + 47T^{2} \)
53 \( 1 + (-0.271 - 0.156i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 3.21T + 59T^{2} \)
61 \( 1 + 3.74T + 61T^{2} \)
67 \( 1 - 7.85iT - 67T^{2} \)
71 \( 1 - 1.55T + 71T^{2} \)
73 \( 1 + (8.33 - 14.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 7.29iT - 79T^{2} \)
83 \( 1 + (-2.25 + 3.89i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (13.5 - 7.84i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.60 + 6.23i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.01178853034097670927092010109, −8.322892138517226163373076086071, −8.037510707129621734971614077368, −6.99526011275467349196075523662, −5.95475627856853494885474470891, −5.17722758469024216333402480380, −4.17831367288554577857375161452, −3.54403819266122750804948922648, −1.39532412887183776850686168274, −0.095005165351283821148356775005, 1.83354540298575469509006592216, 3.07709449770109968519512043961, 4.65773666689779831527923177143, 4.99405209464896130173765968140, 6.30385232636348771017711710066, 6.91127336847774152344848362660, 7.83222674849040777326279996333, 8.557706813531711093166384176344, 9.828599582215341943687704330780, 10.43920458358312773992190932137

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