Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $-0.998 - 0.0576i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.349 + 1.69i)3-s + (1.79 + 3.10i)5-s + (−0.5 + 0.866i)7-s + (−2.75 − 1.18i)9-s + (−1.40 + 2.43i)11-s + (−0.5 − 0.866i)13-s + (−5.89 + 1.95i)15-s − 4.11·17-s + 0.888·19-s + (−1.29 − 1.15i)21-s + (2.93 + 5.08i)23-s + (−3.93 + 6.82i)25-s + (2.97 − 4.25i)27-s + (0.849 − 1.47i)29-s + (−3.49 − 6.05i)31-s + ⋯
L(s)  = 1  + (−0.201 + 0.979i)3-s + (0.802 + 1.38i)5-s + (−0.188 + 0.327i)7-s + (−0.918 − 0.395i)9-s + (−0.423 + 0.733i)11-s + (−0.138 − 0.240i)13-s + (−1.52 + 0.505i)15-s − 0.997·17-s + 0.203·19-s + (−0.282 − 0.251i)21-s + (0.612 + 1.06i)23-s + (−0.787 + 1.36i)25-s + (0.572 − 0.819i)27-s + (0.157 − 0.273i)29-s + (−0.627 − 1.08i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0576i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.998 - 0.0576i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (673, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ -0.998 - 0.0576i)\)
\(L(1)\)  \(\approx\)  \(1.205580170\)
\(L(\frac12)\)  \(\approx\)  \(1.205580170\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.349 - 1.69i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-1.79 - 3.10i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.40 - 2.43i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.11T + 17T^{2} \)
19 \( 1 - 0.888T + 19T^{2} \)
23 \( 1 + (-2.93 - 5.08i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.849 + 1.47i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.49 + 6.05i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.76T + 37T^{2} \)
41 \( 1 + (-2.70 - 4.68i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.60 + 4.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.33 - 2.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.123T + 53T^{2} \)
59 \( 1 + (4.43 + 7.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.93 - 3.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.15 - 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.87T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 + (3.54 - 6.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.05 - 3.56i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.60T + 89T^{2} \)
97 \( 1 + (3.66 - 6.34i)T + (-48.5 - 84.0i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.26519889197550279427806622391, −9.678915520553012944636759093218, −9.123469939661319866744690547038, −7.78659334794629014636363652096, −6.85998347576088283043562044484, −6.02445562218452858072191495753, −5.30216392900746924784244509286, −4.16624888868048008405007585781, −3.01239372225652208495778763560, −2.28294469373731382316932961188, 0.54438454433668087146121881903, 1.66669006410357282136629872329, 2.83495541100542828969457357475, 4.50707828129654069646566518550, 5.30990279334709116258010473616, 6.12569497681904134420797216336, 6.92677190469275569764300527533, 7.986612661485547490310510391965, 8.793496275175378885185686157778, 9.225190043602949312141593724503

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