Properties

Degree $2$
Conductor $672$
Sign $-0.991 - 0.126i$
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 + 2.59i)5-s + (0.5 − 2.59i)7-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s − 4·13-s + 3·15-s + (−2 − 3.46i)17-s + (−2.5 + 0.866i)21-s + (−4 + 6.92i)23-s + (−2 − 3.46i)25-s + 0.999·27-s − 7·29-s + (−5.5 − 9.52i)31-s + (0.499 − 0.866i)33-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.670 + 1.16i)5-s + (0.188 − 0.981i)7-s + (−0.166 + 0.288i)9-s + (0.150 + 0.261i)11-s − 1.10·13-s + 0.774·15-s + (−0.485 − 0.840i)17-s + (−0.545 + 0.188i)21-s + (−0.834 + 1.44i)23-s + (−0.400 − 0.692i)25-s + 0.192·27-s − 1.29·29-s + (−0.987 − 1.71i)31-s + (0.0870 − 0.150i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.991 - 0.126i$
Motivic weight: \(1\)
Character: $\chi_{672} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
good5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7T + 29T^{2} \)
31 \( 1 + (5.5 + 9.52i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.5 - 9.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.5 + 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-3 - 5.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−10.14281054346396365067246227295, −9.401512696234622475758444116524, −7.78153770049909816015557996524, −7.40001818451541070339112726798, −6.84888932591344943018180571278, −5.60390831541413157464302935074, −4.35529245539687864170234320768, −3.37361162170638058414250233083, −2.02994942434856337974335766238, 0, 2.01674172808992553288173504584, 3.63209939211472966796986145547, 4.69026023326786902318418656434, 5.28429918352496554247769353546, 6.34664510997124493103410027810, 7.64519093784188859379067915862, 8.714912023034238691459997068903, 8.901249933039359500505315275459, 10.08584248173242507009655876891

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