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Properties

Label 2-1008-63.5-c1-0-39
Degree $2$
Conductor $1008$
Sign $-0.990 - 0.135i$
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  + (−1.52 + 0.816i)3-s + (−1.82 − 3.15i)5-s + (1.58 − 2.12i)7-s + (1.66 − 2.49i)9-s + (−4.38 − 2.53i)11-s + (2.94 + 1.69i)13-s + (5.35 + 3.33i)15-s + (−0.774 − 1.34i)17-s + (0.707 + 0.408i)19-s + (−0.686 + 4.53i)21-s + (−1.47 + 0.850i)23-s + (−4.13 + 7.17i)25-s + (−0.511 + 5.17i)27-s + (−3.60 + 2.08i)29-s − 2.16i·31-s + ⋯
L(s)  = 1  + (−0.881 + 0.471i)3-s + (−0.814 − 1.41i)5-s + (0.598 − 0.801i)7-s + (0.555 − 0.831i)9-s + (−1.32 − 0.763i)11-s + (0.816 + 0.471i)13-s + (1.38 + 0.860i)15-s + (−0.187 − 0.325i)17-s + (0.162 + 0.0936i)19-s + (−0.149 + 0.988i)21-s + (−0.307 + 0.177i)23-s + (−0.827 + 1.43i)25-s + (−0.0984 + 0.995i)27-s + (−0.669 + 0.386i)29-s − 0.389i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.990 - 0.135i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.990 - 0.135i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3482012759\)
\(L(\frac12)\) \(\approx\) \(0.3482012759\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.52 - 0.816i)T \)
7 \( 1 + (-1.58 + 2.12i)T \)
good5 \( 1 + (1.82 + 3.15i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.38 + 2.53i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.94 - 1.69i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.774 + 1.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.707 - 0.408i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.47 - 0.850i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.60 - 2.08i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.16iT - 31T^{2} \)
37 \( 1 + (3.39 - 5.88i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.01 + 1.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.06 + 5.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.74T + 47T^{2} \)
53 \( 1 + (11.4 - 6.63i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 2.17T + 59T^{2} \)
61 \( 1 + 7.25iT - 61T^{2} \)
67 \( 1 + 2.45T + 67T^{2} \)
71 \( 1 - 6.74iT - 71T^{2} \)
73 \( 1 + (-3.76 + 2.17i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 + (0.768 + 1.33i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.01 - 10.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.59 - 3.23i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.540830656687128947866317919178, −8.610628385798820398843801391499, −7.993445840483166948969298632517, −7.08909277877573271172960804731, −5.77353105306570758214948136007, −5.07928245695589120508770333717, −4.36089303647912191409574920004, −3.55803078186823501153468792284, −1.31862393606902695681812425467, −0.19069735290478271632564933445, 1.97717368222187136662104086433, 2.99229396720460044860995172102, 4.33214321160066730564300292092, 5.39812980401878866907501693257, 6.13723416450958475950716254505, 7.12465876085123135847109226490, 7.73410983043744319123485303682, 8.380560394162158424768981318229, 9.893239904143676402933787697555, 10.77453802281046993341087447192

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