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Properties

Label 2-1008-21.5-c3-0-10
Degree $2$
Conductor $1008$
Sign $-0.874 - 0.485i$
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  + (−3.30 + 5.73i)5-s + (4.31 + 18.0i)7-s + (−1.54 + 0.893i)11-s − 4.72i·13-s + (23.8 + 41.3i)17-s + (40.1 + 23.1i)19-s + (−30.2 − 17.4i)23-s + (40.5 + 70.3i)25-s + 48.3i·29-s + (107. − 62.2i)31-s + (−117. − 34.8i)35-s + (−137. + 238. i)37-s + 37.3·41-s + 215.·43-s + (53.1 − 91.9i)47-s + ⋯
L(s)  = 1  + (−0.295 + 0.512i)5-s + (0.233 + 0.972i)7-s + (−0.0424 + 0.0244i)11-s − 0.100i·13-s + (0.340 + 0.589i)17-s + (0.484 + 0.279i)19-s + (−0.273 − 0.158i)23-s + (0.324 + 0.562i)25-s + 0.309i·29-s + (0.624 − 0.360i)31-s + (−0.567 − 0.168i)35-s + (−0.612 + 1.06i)37-s + 0.142·41-s + 0.765·43-s + (0.164 − 0.285i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.485i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.874 - 0.485i$
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: \((2,\ 1008,\ (\ :3/2),\ -0.874 - 0.485i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.286786281\)
\(L(\frac12)\) \(\approx\) \(1.286786281\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.31 - 18.0i)T \)
good5 \( 1 + (3.30 - 5.73i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (1.54 - 0.893i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 4.72iT - 2.19e3T^{2} \)
17 \( 1 + (-23.8 - 41.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-40.1 - 23.1i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (30.2 + 17.4i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 48.3iT - 2.43e4T^{2} \)
31 \( 1 + (-107. + 62.2i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (137. - 238. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 37.3T + 6.89e4T^{2} \)
43 \( 1 - 215.T + 7.95e4T^{2} \)
47 \( 1 + (-53.1 + 91.9i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (233. - 134. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (149. + 259. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (292. + 168. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-188. - 326. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 816. iT - 3.57e5T^{2} \)
73 \( 1 + (596. - 344. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-307. + 531. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.32e3T + 5.71e5T^{2} \)
89 \( 1 + (480. - 832. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 449. iT - 9.12e5T^{2} \)
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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

−9.944397412995400272961593114345, −9.062753889794020339882687683254, −8.238709610016819796656140785936, −7.52504309563918538347826576429, −6.47890070042712917997871365137, −5.69085702736690483143425006785, −4.76763774714029611854698317889, −3.53671568776723822966086165836, −2.66611329640925517641385066647, −1.42985378586544638513203112773, 0.33914193591318873738982125261, 1.33211228109544939000276061075, 2.83782995889494577532676049472, 4.01670652418798152751128332014, 4.71450728747696838361008757139, 5.70025532604837996698781586250, 6.87796559801147406441257186906, 7.56268318965291582065216554435, 8.338743420367952688499156889720, 9.246628921886140091667686452365

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