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Properties

Label 2-1008-63.4-c1-0-7
Degree $2$
Conductor $1008$
Sign $-0.343 - 0.939i$
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  + (−1.70 − 0.327i)3-s + 0.340·5-s + (−1.09 + 2.40i)7-s + (2.78 + 1.11i)9-s + 0.671·11-s + (1.62 − 2.81i)13-s + (−0.578 − 0.111i)15-s + (−1.10 + 1.90i)17-s + (−0.242 − 0.419i)19-s + (2.65 − 3.73i)21-s − 4.18·23-s − 4.88·25-s + (−4.37 − 2.80i)27-s + (0.478 + 0.829i)29-s + (1.04 + 1.80i)31-s + ⋯
L(s)  = 1  + (−0.982 − 0.188i)3-s + 0.152·5-s + (−0.414 + 0.909i)7-s + (0.928 + 0.370i)9-s + 0.202·11-s + (0.450 − 0.779i)13-s + (−0.149 − 0.0287i)15-s + (−0.266 + 0.462i)17-s + (−0.0555 − 0.0961i)19-s + (0.579 − 0.815i)21-s − 0.873·23-s − 0.976·25-s + (−0.842 − 0.539i)27-s + (0.0889 + 0.154i)29-s + (0.187 + 0.323i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.343 - 0.939i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.343 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6862328686\)
\(L(\frac12)\) \(\approx\) \(0.6862328686\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 + 0.327i)T \)
7 \( 1 + (1.09 - 2.40i)T \)
good5 \( 1 - 0.340T + 5T^{2} \)
11 \( 1 - 0.671T + 11T^{2} \)
13 \( 1 + (-1.62 + 2.81i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.10 - 1.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.242 + 0.419i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.18T + 23T^{2} \)
29 \( 1 + (-0.478 - 0.829i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.04 - 1.80i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.81 - 8.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.90 - 6.75i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.66 - 6.34i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.34 - 2.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.12 - 10.6i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.47 + 4.28i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.76 + 3.04i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.16 - 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.57T + 71T^{2} \)
73 \( 1 + (3.71 - 6.43i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.00 - 8.67i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.47 + 4.28i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.52 + 14.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.23 - 7.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.12859300396945604574596109282, −9.652350029074562608942896329891, −8.468850598614363231784286207873, −7.75333551352606144721420521355, −6.42394388887854635560271774254, −6.10841808941071840767869909347, −5.21627330477646844180168813300, −4.15494843084277937383055967295, −2.82060986091606835654589328913, −1.44044912537402324783629642329, 0.36785108269072483201272225819, 1.89559769770546302926299549815, 3.75787119171499527830354876272, 4.28074435534109196788291186050, 5.48053870816497855759590799630, 6.33056945940089205471720562734, 6.97738031716272642202638661367, 7.86381378043182922768357524447, 9.181263328230520044659184555041, 9.802828807686772248476409170262

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