Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $-0.0832 - 0.996i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.933 − 1.45i)3-s + (−1.23 + 2.13i)5-s + (−0.5 − 0.866i)7-s + (−1.25 − 2.72i)9-s + (2.32 + 4.02i)11-s + (−3.55 + 6.15i)13-s + (1.96 + 3.78i)15-s − 4.51·17-s − 4.32·19-s + (−1.73 − 0.0789i)21-s + (2.93 − 5.08i)23-s + (−0.527 − 0.912i)25-s + (−5.14 − 0.708i)27-s + (3.48 + 6.04i)29-s + (−3.69 + 6.39i)31-s + ⋯
L(s)  = 1  + (0.538 − 0.842i)3-s + (−0.550 + 0.952i)5-s + (−0.188 − 0.327i)7-s + (−0.419 − 0.907i)9-s + (0.700 + 1.21i)11-s + (−0.985 + 1.70i)13-s + (0.506 + 0.977i)15-s − 1.09·17-s − 0.992·19-s + (−0.377 − 0.0172i)21-s + (0.611 − 1.05i)23-s + (−0.105 − 0.182i)25-s + (−0.990 − 0.136i)27-s + (0.647 + 1.12i)29-s + (−0.662 + 1.14i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0832 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0832 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.0832 - 0.996i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (337, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ -0.0832 - 0.996i)\)
\(L(1)\)  \(\approx\)  \(1.005167100\)
\(L(\frac12)\)  \(\approx\)  \(1.005167100\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.933 + 1.45i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (1.23 - 2.13i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.32 - 4.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.55 - 6.15i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.51T + 17T^{2} \)
19 \( 1 + 4.32T + 19T^{2} \)
23 \( 1 + (-2.93 + 5.08i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.48 - 6.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.69 - 6.39i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.726T + 37T^{2} \)
41 \( 1 + (0.136 - 0.236i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.41 + 4.18i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.83 - 3.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.05T + 53T^{2} \)
59 \( 1 + (-4.56 + 7.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.90 - 11.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.663 - 1.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 + 4.32T + 73T^{2} \)
79 \( 1 + (-3.21 - 5.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.742 - 1.28i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.83T + 89T^{2} \)
97 \( 1 + (-0.246 - 0.426i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.19628576573309304785478519347, −9.105374679360338258680061324168, −8.666568299671008844981208028562, −7.20491689322588801442120698906, −6.93670597072540966677283785116, −6.60255952326655432965003884455, −4.66893205225040431710220052693, −3.92161640037118546323911048747, −2.65689305551791856149339773300, −1.79749541168992469916997131788, 0.40774962780391401145471426510, 2.45797617606362914769964708672, 3.50130075459037375592109236441, 4.38947741893270277964301036358, 5.25086164311522622186102404553, 6.09548684183916287361136146872, 7.56674146338922983913008076537, 8.363105850909969329428678567778, 8.811922099553362245720731497925, 9.605734919808340039347473429599

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