Properties

Label 2-1008-21.17-c3-0-38
Degree $2$
Conductor $1008$
Sign $-0.387 + 0.921i$
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  + (−9.74 − 16.8i)5-s + (18.4 − 1.67i)7-s + (15.1 + 8.72i)11-s − 16.5i·13-s + (68.6 − 118. i)17-s + (20.6 − 11.9i)19-s + (108. − 62.8i)23-s + (−127. + 220. i)25-s + 105. i·29-s + (95.0 + 54.8i)31-s + (−207. − 294. i)35-s + (58.5 + 101. i)37-s + 348.·41-s + 141.·43-s + (−172. − 299. i)47-s + ⋯
L(s)  = 1  + (−0.871 − 1.50i)5-s + (0.995 − 0.0902i)7-s + (0.414 + 0.239i)11-s − 0.352i·13-s + (0.979 − 1.69i)17-s + (0.249 − 0.144i)19-s + (0.987 − 0.570i)23-s + (−1.01 + 1.76i)25-s + 0.673i·29-s + (0.550 + 0.318i)31-s + (−1.00 − 1.42i)35-s + (0.260 + 0.450i)37-s + 1.32·41-s + 0.503·43-s + (−0.536 − 0.929i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.387 + 0.921i$
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: \((2,\ 1008,\ (\ :3/2),\ -0.387 + 0.921i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.078639331\)
\(L(\frac12)\) \(\approx\) \(2.078639331\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.4 + 1.67i)T \)
good5 \( 1 + (9.74 + 16.8i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-15.1 - 8.72i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 16.5iT - 2.19e3T^{2} \)
17 \( 1 + (-68.6 + 118. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-20.6 + 11.9i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-108. + 62.8i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 105. iT - 2.43e4T^{2} \)
31 \( 1 + (-95.0 - 54.8i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-58.5 - 101. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 348.T + 6.89e4T^{2} \)
43 \( 1 - 141.T + 7.95e4T^{2} \)
47 \( 1 + (172. + 299. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-149. - 86.1i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-297. + 515. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (561. - 324. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-64.9 + 112. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 908. iT - 3.57e5T^{2} \)
73 \( 1 + (44.8 + 25.9i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (474. + 822. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.30e3T + 5.71e5T^{2} \)
89 \( 1 + (210. + 364. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 553. iT - 9.12e5T^{2} \)
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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

−9.089526707505043863304790686919, −8.552102591786979265009238989567, −7.70195823345726235736783573763, −7.12167976308807408653217264408, −5.48926032767211654623330462751, −4.89109905215752336195503037300, −4.26339411748107880878088903083, −2.96401995201071206609107053114, −1.28964411214958552052947750483, −0.63578945973081840651196700084, 1.24356314283796044322730426051, 2.57839358916078353764797216014, 3.63066234095618383364351506274, 4.31095042865317815512692425560, 5.72728084803340618839450543834, 6.47643531150801816757110774011, 7.59590884495377197523980730763, 7.84563484923947399966544069777, 8.910936099509994473114500090198, 10.05360669603224092783840118543

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