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Properties

Label 2-1008-21.17-c3-0-30
Degree $2$
Conductor $1008$
Sign $0.576 + 0.816i$
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  + (−5.92 − 10.2i)5-s + (18.3 + 2.35i)7-s + (22.9 + 13.2i)11-s + 1.60i·13-s + (42.1 − 73.0i)17-s + (89.8 − 51.8i)19-s + (−168. + 97.4i)23-s + (−7.62 + 13.2i)25-s − 28.8i·29-s + (155. + 89.9i)31-s + (−84.6 − 202. i)35-s + (172. + 299. i)37-s + 18.1·41-s − 244.·43-s + (103. + 178. i)47-s + ⋯
L(s)  = 1  + (−0.529 − 0.917i)5-s + (0.991 + 0.127i)7-s + (0.630 + 0.363i)11-s + 0.0343i·13-s + (0.601 − 1.04i)17-s + (1.08 − 0.626i)19-s + (−1.53 + 0.883i)23-s + (−0.0610 + 0.105i)25-s − 0.184i·29-s + (0.902 + 0.521i)31-s + (−0.408 − 0.977i)35-s + (0.767 + 1.32i)37-s + 0.0691·41-s − 0.868·43-s + (0.320 + 0.554i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 + 0.816i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.576 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.576 + 0.816i$
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: \((2,\ 1008,\ (\ :3/2),\ 0.576 + 0.816i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.264153856\)
\(L(\frac12)\) \(\approx\) \(2.264153856\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.3 - 2.35i)T \)
good5 \( 1 + (5.92 + 10.2i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-22.9 - 13.2i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 1.60iT - 2.19e3T^{2} \)
17 \( 1 + (-42.1 + 73.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-89.8 + 51.8i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (168. - 97.4i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 28.8iT - 2.43e4T^{2} \)
31 \( 1 + (-155. - 89.9i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-172. - 299. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 18.1T + 6.89e4T^{2} \)
43 \( 1 + 244.T + 7.95e4T^{2} \)
47 \( 1 + (-103. - 178. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-67.7 - 39.1i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (43.7 - 75.7i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-704. + 406. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (242. - 419. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 950. iT - 3.57e5T^{2} \)
73 \( 1 + (712. + 411. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (211. + 366. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 635.T + 5.71e5T^{2} \)
89 \( 1 + (724. + 1.25e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.31e3iT - 9.12e5T^{2} \)
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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

−9.443805158282363708612308041674, −8.529110560202952859124316796018, −7.88016635315443521925090028017, −7.15219000642833488105225094025, −5.87356936747215133965361150814, −4.88551126151790275375006375203, −4.41436132298367824559458823567, −3.12881477956332157668008566135, −1.65777160492374171780537148144, −0.71772195396185377465441802682, 1.01566117163863701868143806615, 2.26422593973083115350221941033, 3.57983737122434952552578865600, 4.17242404519013004835956561946, 5.51169684154093360317485964505, 6.30244411511624037775757695440, 7.34567691527584857223964915778, 7.982714264341720071429883879330, 8.659657634169918217672342414780, 9.967644371936562377557379129394

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