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Properties

Label 2-1008-63.59-c1-0-44
Degree $2$
Conductor $1008$
Sign $-0.905 - 0.423i$
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  + (−0.361 − 1.69i)3-s − 0.900·5-s + (−1.05 − 2.42i)7-s + (−2.73 + 1.22i)9-s − 3.12i·11-s + (1.99 + 1.14i)13-s + (0.325 + 1.52i)15-s + (2.57 − 4.46i)17-s + (−2.38 + 1.37i)19-s + (−3.72 + 2.66i)21-s + 1.71i·23-s − 4.18·25-s + (3.06 + 4.19i)27-s + (−1.85 + 1.07i)29-s + (−8.66 + 5.00i)31-s + ⋯
L(s)  = 1  + (−0.208 − 0.977i)3-s − 0.402·5-s + (−0.399 − 0.916i)7-s + (−0.912 + 0.408i)9-s − 0.943i·11-s + (0.552 + 0.318i)13-s + (0.0840 + 0.393i)15-s + (0.624 − 1.08i)17-s + (−0.546 + 0.315i)19-s + (−0.813 + 0.582i)21-s + 0.357i·23-s − 0.837·25-s + (0.589 + 0.807i)27-s + (−0.344 + 0.198i)29-s + (−1.55 + 0.898i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 - 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.905 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.905 - 0.423i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.905 - 0.423i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5153835733\)
\(L(\frac12)\) \(\approx\) \(0.5153835733\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.361 + 1.69i)T \)
7 \( 1 + (1.05 + 2.42i)T \)
good5 \( 1 + 0.900T + 5T^{2} \)
11 \( 1 + 3.12iT - 11T^{2} \)
13 \( 1 + (-1.99 - 1.14i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.57 + 4.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.38 - 1.37i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.71iT - 23T^{2} \)
29 \( 1 + (1.85 - 1.07i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.66 - 5.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.73 + 8.20i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.22 - 2.11i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.273 - 0.473i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.93 - 6.80i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-12.0 - 6.97i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.99 + 6.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.28 - 3.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.83 - 3.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 + (10.9 + 6.30i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.27 - 5.67i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.184 - 0.319i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.00 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.86 - 5.12i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.361878628526690691575509727989, −8.547907985556225255109058202091, −7.57977423379297751333823031328, −7.14263233392248998292521887672, −6.14190501449364163457916112533, −5.37339286353388065649732234214, −3.92721297073681447843942498742, −3.11831593765259594234852821768, −1.53610559016826828270135673626, −0.23933500260935710587043960458, 2.13118556786636357034697960012, 3.47062289842439872762669292604, 4.14736531046957261348150548124, 5.33617902262053264922019114746, 5.93375734454920076460044367318, 6.99461109955420921091335101068, 8.261939397379523378202859274011, 8.755022169346718595949994346832, 9.794712805095502600927746717001, 10.22675373540734404039090758433

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