Properties

Degree $2$
Conductor $1008$
Sign $0.939 + 0.342i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·3-s + (1.5 − 2.59i)5-s + (0.5 + 0.866i)7-s − 2.99·9-s + (−3 − 5.19i)11-s + (−1 + 1.73i)13-s + (4.5 + 2.59i)15-s + 6·17-s + 7·19-s + (−1.49 + 0.866i)21-s + (1.5 − 2.59i)23-s + (−2 − 3.46i)25-s − 5.19i·27-s + (−3 − 5.19i)29-s + (1 − 1.73i)31-s + ⋯
L(s)  = 1  + 0.999i·3-s + (0.670 − 1.16i)5-s + (0.188 + 0.327i)7-s − 0.999·9-s + (−0.904 − 1.56i)11-s + (−0.277 + 0.480i)13-s + (1.16 + 0.670i)15-s + 1.45·17-s + 1.60·19-s + (−0.327 + 0.188i)21-s + (0.312 − 0.541i)23-s + (−0.400 − 0.692i)25-s − 0.999i·27-s + (−0.557 − 0.964i)29-s + (0.179 − 0.311i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.939 + 0.342i$
Motivic weight: \(1\)
Character: $\chi_{1008} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.716254458\)
\(L(\frac12)\) \(\approx\) \(1.716254458\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1 - 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.673652374575824079255213528516, −9.307061164784269253296220964657, −8.377156827410472067792029340427, −7.79846340776504668548893833396, −6.01251393840904466698967811897, −5.43571162268061002545060068987, −4.94992987742234321206390062877, −3.63935879178098036372027016555, −2.63813215317811622380824500758, −0.872297971572014747466251686579, 1.39981180678212039421183646151, 2.55325418531545072069682681972, 3.30663373907994560975155351255, 5.15389003042644954330046899002, 5.69234507888719946771370170362, 6.97158293428436286731633916590, 7.36426437142942125919744457445, 7.903983053165542946697583603814, 9.383947140078458358520778779524, 10.10988741422847538334413575735

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