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Properties

Label 2-1008-21.5-c1-0-8
Degree $2$
Conductor $1008$
Sign $0.851 + 0.524i$
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  + (−1.22 + 2.12i)5-s + (−0.5 − 2.59i)7-s + (3.67 − 2.12i)11-s − 1.73i·13-s + (2.44 + 4.24i)17-s + (−4.5 − 2.59i)19-s + (−0.499 − 0.866i)25-s − 8.48i·29-s + (1.5 − 0.866i)31-s + (6.12 + 2.12i)35-s + (2.5 − 4.33i)37-s + 12.2·41-s + 11·43-s + (1.22 − 2.12i)47-s + (−6.5 + 2.59i)49-s + ⋯
L(s)  = 1  + (−0.547 + 0.948i)5-s + (−0.188 − 0.981i)7-s + (1.10 − 0.639i)11-s − 0.480i·13-s + (0.594 + 1.02i)17-s + (−1.03 − 0.596i)19-s + (−0.0999 − 0.173i)25-s − 1.57i·29-s + (0.269 − 0.155i)31-s + (1.03 + 0.358i)35-s + (0.410 − 0.711i)37-s + 1.91·41-s + 1.67·43-s + (0.178 − 0.309i)47-s + (−0.928 + 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.851 + 0.524i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.851 + 0.524i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.421458909\)
\(L(\frac12)\) \(\approx\) \(1.421458909\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (0.5 + 2.59i)T \)
good5 \( 1 + (1.22 - 2.12i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.67 + 2.12i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 + (-2.44 - 4.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.5 + 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.48iT - 29T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.5 + 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 12.2T + 41T^{2} \)
43 \( 1 - 11T + 43T^{2} \)
47 \( 1 + (-1.22 + 2.12i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.34 + 4.24i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.44 - 4.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 + 1.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.7iT - 71T^{2} \)
73 \( 1 + (-13.5 + 7.79i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 + (7.34 - 12.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.46iT - 97T^{2} \)
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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

−10.03569742367465948800138956421, −9.093017873004279340573295513850, −8.042997350450154299072296328853, −7.40988005696795121764358459280, −6.49733477223224894127395036635, −5.89150992363185737415777815755, −4.13250126396295543123038199517, −3.82439955193218684966844060004, −2.58463311752336820137407450943, −0.78701915851157759413627843019, 1.21039036892824810611887470924, 2.58307117411532439909925219121, 3.98207949557807828697298084418, 4.69624668658080745149190814630, 5.69796842834653934276267755585, 6.64209921512933691214298346026, 7.58946420052212785667005927621, 8.605810478131454890352475924168, 9.108934458645339915034566193776, 9.747787325171538946622918569539

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