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Properties

Label 2-1008-21.5-c3-0-29
Degree $2$
Conductor $1008$
Sign $0.791 + 0.611i$
Analytic cond. $59.4739$
Root an. cond. $7.71193$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.00 − 8.66i)5-s + (−1.56 − 18.4i)7-s + (8.94 − 5.16i)11-s + 52.4i·13-s + (−0.584 − 1.01i)17-s + (86.7 + 50.0i)19-s + (90.1 + 52.0i)23-s + (12.4 + 21.6i)25-s + 187. i·29-s + (107. − 62.0i)31-s + (−167. − 78.6i)35-s + (16.0 − 27.7i)37-s + 415.·41-s + 193.·43-s + (196. − 340. i)47-s + ⋯
L(s)  = 1  + (0.447 − 0.774i)5-s + (−0.0847 − 0.996i)7-s + (0.245 − 0.141i)11-s + 1.11i·13-s + (−0.00833 − 0.0144i)17-s + (1.04 + 0.604i)19-s + (0.817 + 0.471i)23-s + (0.0999 + 0.173i)25-s + 1.20i·29-s + (0.622 − 0.359i)31-s + (−0.809 − 0.379i)35-s + (0.0712 − 0.123i)37-s + 1.58·41-s + 0.685·43-s + (0.609 − 1.05i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 + 0.611i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.791 + 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.791 + 0.611i$
Analytic conductor: \(59.4739\)
Root analytic conductor: \(7.71193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :3/2),\ 0.791 + 0.611i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.477522352\)
\(L(\frac12)\) \(\approx\) \(2.477522352\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.56 + 18.4i)T \)
good5 \( 1 + (-5.00 + 8.66i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-8.94 + 5.16i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 52.4iT - 2.19e3T^{2} \)
17 \( 1 + (0.584 + 1.01i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-86.7 - 50.0i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-90.1 - 52.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 187. iT - 2.43e4T^{2} \)
31 \( 1 + (-107. + 62.0i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-16.0 + 27.7i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 415.T + 6.89e4T^{2} \)
43 \( 1 - 193.T + 7.95e4T^{2} \)
47 \( 1 + (-196. + 340. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-74.7 + 43.1i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (102. + 177. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (183. + 105. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (364. + 631. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 315. iT - 3.57e5T^{2} \)
73 \( 1 + (899. - 519. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-607. + 1.05e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 333.T + 5.71e5T^{2} \)
89 \( 1 + (-168. + 291. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 893. iT - 9.12e5T^{2} \)
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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

−9.312222340896080894441969613707, −8.972659681355256641917924353434, −7.69788200410704950577545433844, −7.06675065531573699452680887681, −6.05627111769422550223168084498, −5.07634370647730758308288479150, −4.24618923766747110807149710105, −3.25005929959398953768220469797, −1.66232899576654306177293220547, −0.848884544881594831078630071344, 0.916217186727720811780417071942, 2.58735444331409828457061480877, 2.91749691834565578172816442367, 4.45119783321462120636396839780, 5.56894087766163599330226433568, 6.13875366080370857754800938431, 7.11849396795618570328344653461, 7.980617422928020941143906743121, 8.970101982390618395370991580898, 9.646453966829872367014640469960

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