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Properties

Label 2-1008-336.251-c0-0-1
Degree $2$
Conductor $1008$
Sign $0.845 - 0.533i$
Analytic cond. $0.503057$
Root an. cond. $0.709265$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + 7-s + (0.707 + 0.707i)8-s + (−0.707 + 0.707i)14-s − 1.00·16-s − 1.41i·23-s − i·25-s − 1.00i·28-s + (1.41 + 1.41i)29-s + (0.707 − 0.707i)32-s + (1 + i)37-s + (−1 + i)43-s + (1.00 + 1.00i)46-s + 49-s + (0.707 + 0.707i)50-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + 7-s + (0.707 + 0.707i)8-s + (−0.707 + 0.707i)14-s − 1.00·16-s − 1.41i·23-s − i·25-s − 1.00i·28-s + (1.41 + 1.41i)29-s + (0.707 − 0.707i)32-s + (1 + i)37-s + (−1 + i)43-s + (1.00 + 1.00i)46-s + 49-s + (0.707 + 0.707i)50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.845 - 0.533i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :0),\ 0.845 - 0.533i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8095461940\)
\(L(\frac12)\) \(\approx\) \(0.8095461940\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + iT^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (1 - i)T - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 2iT - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.27264338832845046662051164311, −9.241939670125630611461851137052, −8.329885671463131226492144601556, −8.043589387327027405704179864481, −6.85328516660309158392607666855, −6.24317707670457381859274887728, −5.01796363828645461043914450467, −4.48987638737463982146214976482, −2.63406787529765031827056095604, −1.27796914641113696921345692018, 1.33917560318477689524177416036, 2.46162918426492289141921853941, 3.69853713462040125158001461349, 4.64302187759125211287668400706, 5.74750338617783379498960264269, 7.09691545257616999729711077230, 7.77701328485827496623586933000, 8.496852000437663490132188281331, 9.329879049027684116715564634712, 10.09643209550860033961594003695

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