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Properties

Label 2-1008-63.59-c1-0-7
Degree $2$
Conductor $1008$
Sign $-0.967 - 0.254i$
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  + (−0.149 + 1.72i)3-s − 2.22·5-s + (2.45 + 0.996i)7-s + (−2.95 − 0.514i)9-s − 1.17i·11-s + (3.12 + 1.80i)13-s + (0.331 − 3.83i)15-s + (−3.71 + 6.42i)17-s + (3.05 − 1.76i)19-s + (−2.08 + 4.08i)21-s + 5.81i·23-s − 0.0674·25-s + (1.32 − 5.02i)27-s + (−6.04 + 3.48i)29-s + (−6.88 + 3.97i)31-s + ⋯
L(s)  = 1  + (−0.0860 + 0.996i)3-s − 0.993·5-s + (0.926 + 0.376i)7-s + (−0.985 − 0.171i)9-s − 0.353i·11-s + (0.866 + 0.500i)13-s + (0.0854 − 0.989i)15-s + (−0.900 + 1.55i)17-s + (0.700 − 0.404i)19-s + (−0.454 + 0.890i)21-s + 1.21i·23-s − 0.0134·25-s + (0.255 − 0.966i)27-s + (−1.12 + 0.648i)29-s + (−1.23 + 0.713i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.967 - 0.254i$
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: \((2,\ 1008,\ (\ :1/2),\ -0.967 - 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8575144937\)
\(L(\frac12)\) \(\approx\) \(0.8575144937\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.149 - 1.72i)T \)
7 \( 1 + (-2.45 - 0.996i)T \)
good5 \( 1 + 2.22T + 5T^{2} \)
11 \( 1 + 1.17iT - 11T^{2} \)
13 \( 1 + (-3.12 - 1.80i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.71 - 6.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.05 + 1.76i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.81iT - 23T^{2} \)
29 \( 1 + (6.04 - 3.48i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.88 - 3.97i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.54 + 9.60i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.809 - 1.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.904 + 1.56i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.26 + 7.38i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.62 + 5.55i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.00 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.09 + 4.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.96 - 8.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.67iT - 71T^{2} \)
73 \( 1 + (-6.92 - 3.99i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.25 - 3.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.390 - 0.677i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.75 - 3.03i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.49 - 2.01i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.66832264223330610311975015113, −9.286402386180649825666546817746, −8.755793441910081516165906911000, −8.086737754789827060361832439106, −7.08645930719833581656914683178, −5.78184991957245559056577932193, −5.13986233287009127761284949775, −3.87579068122420486445150306842, −3.65120603330962493216221761790, −1.81692451824457066164775895546, 0.39613130475574683293804641822, 1.76701011278852526966110454673, 3.10105074410835911401567530962, 4.29109045221339311919326568907, 5.21213891544255378363613797361, 6.32640795174465166069615511906, 7.34819291724109941452656854310, 7.73655689834967120619773417085, 8.471606647939042867298184155170, 9.407201151394948723485035422465

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