Properties

Label 2-1008-21.17-c3-0-22
Degree $2$
Conductor $1008$
Sign $0.637 - 0.770i$
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  + (6.38 + 11.0i)5-s + (−2.53 − 18.3i)7-s + (46.8 + 27.0i)11-s + 8.85i·13-s + (−34.4 + 59.6i)17-s + (141. − 81.9i)19-s + (−81.3 + 46.9i)23-s + (−18.9 + 32.8i)25-s + 119. i·29-s + (−85.6 − 49.4i)31-s + (186. − 145. i)35-s + (−47.0 − 81.5i)37-s + 259.·41-s − 5.01·43-s + (−28.6 − 49.6i)47-s + ⋯
L(s)  = 1  + (0.570 + 0.988i)5-s + (−0.137 − 0.990i)7-s + (1.28 + 0.741i)11-s + 0.188i·13-s + (−0.491 + 0.851i)17-s + (1.71 − 0.989i)19-s + (−0.737 + 0.425i)23-s + (−0.151 + 0.262i)25-s + 0.765i·29-s + (−0.496 − 0.286i)31-s + (0.901 − 0.700i)35-s + (−0.209 − 0.362i)37-s + 0.987·41-s − 0.0177·43-s + (−0.0889 − 0.154i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.637 - 0.770i$
Analytic conductor: \(59.4739\)
Root analytic conductor: \(7.71193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :3/2),\ 0.637 - 0.770i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.525676763\)
\(L(\frac12)\) \(\approx\) \(2.525676763\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.53 + 18.3i)T \)
good5 \( 1 + (-6.38 - 11.0i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-46.8 - 27.0i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 8.85iT - 2.19e3T^{2} \)
17 \( 1 + (34.4 - 59.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-141. + 81.9i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (81.3 - 46.9i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 119. iT - 2.43e4T^{2} \)
31 \( 1 + (85.6 + 49.4i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (47.0 + 81.5i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 259.T + 6.89e4T^{2} \)
43 \( 1 + 5.01T + 7.95e4T^{2} \)
47 \( 1 + (28.6 + 49.6i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-407. - 235. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-112. + 195. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-370. + 213. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-81.9 + 141. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 79.8iT - 3.57e5T^{2} \)
73 \( 1 + (-666. - 384. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-267. - 463. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 438.T + 5.71e5T^{2} \)
89 \( 1 + (-12.8 - 22.2i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.38e3iT - 9.12e5T^{2} \)
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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

−9.656439819878640892310523975720, −9.203797855849677409990674930771, −7.81339824163464863673302758507, −6.91180168675228060935348614734, −6.65626691106585597751184782079, −5.47163045428666195810168130888, −4.21202557833419635801408804251, −3.49092431163315519302533645381, −2.21306730694804579159854159416, −1.08155147237021322342472759024, 0.74192405137990296685819672562, 1.77989376435194340466176270890, 3.04881101787818497175047768596, 4.18271934248755108334327831506, 5.40399860979940814906275966528, 5.77528916991061827666544988002, 6.80225735462509141496770741374, 8.024461383624443351538014414706, 8.816576606144786800187948832249, 9.370028002739207280656140272249

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