Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 7 $
Sign $-0.895 + 0.444i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (−1.5 + 0.866i)5-s + (−2.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (−4.5 − 2.59i)11-s − 6.92i·13-s − 1.73i·15-s + (3 + 1.73i)17-s + (1 + 1.73i)19-s + (0.500 − 2.59i)21-s + (−6 + 3.46i)23-s + (−1 + 1.73i)25-s + 0.999·27-s − 9·29-s + (0.5 − 0.866i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.670 + 0.387i)5-s + (−0.944 + 0.327i)7-s + (−0.166 − 0.288i)9-s + (−1.35 − 0.783i)11-s − 1.92i·13-s − 0.447i·15-s + (0.727 + 0.420i)17-s + (0.229 + 0.397i)19-s + (0.109 − 0.566i)21-s + (−1.25 + 0.722i)23-s + (−0.200 + 0.346i)25-s + 0.192·27-s − 1.67·29-s + (0.0898 − 0.155i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $-0.895 + 0.444i$
motivic weight  =  \(1\)
character  :  $\chi_{336} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  1
Selberg data  =  $(2,\ 336,\ (\ :1/2),\ -0.895 + 0.444i)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.5 - 0.866i)T \)
good5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.5 + 2.59i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.92iT - 13T^{2} \)
17 \( 1 + (-3 - 1.73i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6 - 3.46i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6 - 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.92iT - 71T^{2} \)
73 \( 1 + (-6 - 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 15T + 83T^{2} \)
89 \( 1 + (9 - 5.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.66iT - 97T^{2} \)
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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

−10.95998550160901359027347475738, −10.34631137490228176771122800968, −9.545257984635324886593211677281, −8.079031798596557269617026778079, −7.62653990032028339360052445859, −5.85563143382967819386074800199, −5.53124706110679993099122855047, −3.64276965826824894169972705290, −3.02232996053267043969500318282, 0, 2.20627190757909835975655245146, 3.87684834127316412962556989560, 4.94594415883703585734042969171, 6.28642125847517367664409703856, 7.24645603807802844624133882838, 7.934743136864111534373675017690, 9.259414738087848330733795032145, 10.04201671945893653801052312456, 11.18605671006109212954167084842

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