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 + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s − 2.99·9-s + (−3 − 5.19i)11-s + (−3 + 5.19i)13-s + (−1.49 − 0.866i)15-s − 2·17-s − 7·19-s + (1.49 − 0.866i)21-s + (−0.5 + 0.866i)23-s + (2 + 3.46i)25-s + 5.19i·27-s + (−1 − 1.73i)29-s + (5 − 8.66i)31-s + ⋯
L(s)  = 1  − 0.999i·3-s + (0.223 − 0.387i)5-s + (0.188 + 0.327i)7-s − 0.999·9-s + (−0.904 − 1.56i)11-s + (−0.832 + 1.44i)13-s + (−0.387 − 0.223i)15-s − 0.485·17-s − 1.60·19-s + (0.327 − 0.188i)21-s + (−0.104 + 0.180i)23-s + (0.400 + 0.692i)25-s + 0.999i·27-s + (−0.185 − 0.321i)29-s + (0.898 − 1.55i)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

\(L(1)\) \(\approx\) \(0.4810051584\)
\(L(\frac12)\) \(\approx\) \(0.4810051584\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5 + 8.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (-4 + 6.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5 + 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 4T + 89T^{2} \)
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

−9.117398423675225066619430229700, −8.709501614789002665773507504302, −7.890511484569878029122806067196, −6.94412296329149120012633734727, −6.08688729004508515663104924858, −5.38471656937120029026824900334, −4.20506971988925797770657569798, −2.68656046954670565157106998634, −1.89066272747593138913215682996, −0.19802837887515089557502944622, 2.27866574135177906016483310826, 3.11584046593279726622395732545, 4.67022628736874351729717458984, 4.79050826078438600298430538030, 6.10039384850856176607967336767, 7.10000505504824821251188717567, 8.019502367706033541894703541010, 8.796483284033356513921534811987, 9.975751279554894950698185171397, 10.41334741536988331336256046221

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