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Properties

Label 2-1008-336.251-c0-0-2
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 + (−1.41 − 1.41i)11-s + (0.707 + 0.707i)14-s − 1.00·16-s + 2.00i·22-s − 1.41i·23-s − i·25-s − 1.00i·28-s + (0.707 + 0.707i)32-s + (−1 − i)37-s + (1 − i)43-s + (1.41 − 1.41i)44-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s − 7-s + (0.707 − 0.707i)8-s + (−1.41 − 1.41i)11-s + (0.707 + 0.707i)14-s − 1.00·16-s + 2.00i·22-s − 1.41i·23-s − i·25-s − 1.00i·28-s + (0.707 + 0.707i)32-s + (−1 − i)37-s + (1 − i)43-s + (1.41 − 1.41i)44-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

\(L(\frac{1}{2})\) \(\approx\) \(0.3902079261\)
\(L(\frac12)\) \(\approx\) \(0.3902079261\)
\(L(1)\) 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 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 + iT^{2} \)
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 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
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

−9.992029224022607806876411757713, −8.955710391446605331148005050932, −8.413703804688062162251752679680, −7.54558534410578186510715399038, −6.53393257670786953095025793407, −5.59881736891634783717253472005, −4.22818180250713804894757019706, −3.12739575221930658724098176350, −2.47081237265007195870944313653, −0.43856352917825872593213059740, 1.83238480521911020978804287025, 3.16639631267505621644183215246, 4.70932527486936189132079359719, 5.46950382883319850503163999699, 6.43392863888571924741407028823, 7.36964659337501254014753475881, 7.74077888980659572573411408684, 8.955733962302488474664306226350, 9.705581027835523516717913017416, 10.13662797308849714955710890862

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