Properties

Degree $2$
Conductor $672$
Sign $-0.519 + 0.854i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.227 + 0.393i)5-s + (−2.16 − 1.51i)7-s + (−0.499 + 0.866i)9-s + (−2.89 − 5.01i)11-s − 5.88·13-s − 0.454·15-s + (−1.45 − 2.51i)17-s + (2.94 − 5.09i)19-s + (0.227 − 2.63i)21-s + (−1.45 + 2.51i)23-s + (2.39 + 4.15i)25-s − 0.999·27-s + 3.54·29-s + (−2.16 − 3.75i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.101 + 0.176i)5-s + (−0.819 − 0.572i)7-s + (−0.166 + 0.288i)9-s + (−0.873 − 1.51i)11-s − 1.63·13-s − 0.117·15-s + (−0.352 − 0.611i)17-s + (0.674 − 1.16i)19-s + (0.0496 − 0.575i)21-s + (−0.303 + 0.525i)23-s + (0.479 + 0.830i)25-s − 0.192·27-s + 0.658·29-s + (−0.389 − 0.674i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.519 + 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.519 + 0.854i$
Motivic weight: \(1\)
Character: $\chi_{672} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.519 + 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.277452 - 0.493410i\)
\(L(\frac12)\) \(\approx\) \(0.277452 - 0.493410i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.16 + 1.51i)T \)
good5 \( 1 + (0.227 - 0.393i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.89 + 5.01i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.88T + 13T^{2} \)
17 \( 1 + (1.45 + 2.51i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.94 + 5.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.45 - 2.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.54T + 29T^{2} \)
31 \( 1 + (2.16 + 3.75i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.85 - 6.67i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.58T + 41T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 + (-2.45 + 4.25i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.56 + 11.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.896 + 1.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.33 - 4.04i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.94 - 6.82i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.909T + 71T^{2} \)
73 \( 1 + (2.60 + 4.50i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.37 - 2.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.97T + 83T^{2} \)
89 \( 1 + (-2.45 + 4.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.79T + 97T^{2} \)
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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

−10.05194017823937756765190181160, −9.544725968353378747983432767066, −8.561781856354063600440566207347, −7.52128440111570753510541778477, −6.83838496728137281198930651091, −5.51714148138722495166841102735, −4.71810992125832883959134260886, −3.31007562640876040853001415032, −2.74936477547043484423228849895, −0.26672962071162906737515845062, 2.03570098204310812243980361832, 2.88101373773014752042247051757, 4.38883090686921638903833796266, 5.36461961511056220078084620852, 6.49356773281556600155186962534, 7.36330317063398193555302639629, 8.039536167338424272170545868018, 9.156808512847583832541708069585, 9.906287192575718822879900593885, 10.51669272357083574482537157535

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