Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.566 + 0.824i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 + 1.36i)2-s + (0.448 − 1.67i)3-s + (−1.73 + i)4-s + (−2.17 − 4.50i)5-s + 2.44·6-s + (−6.19 + 3.26i)7-s + (−2 − 1.99i)8-s + (−2.59 − 1.50i)9-s + (5.35 − 4.62i)10-s + (−0.187 − 0.325i)11-s + (0.896 + 3.34i)12-s + (−9.32 − 9.32i)13-s + (−6.72 − 7.26i)14-s + (−8.50 + 1.62i)15-s + (1.99 − 3.46i)16-s + (−29.7 − 7.97i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (0.149 − 0.557i)3-s + (−0.433 + 0.250i)4-s + (−0.435 − 0.900i)5-s + 0.408·6-s + (−0.884 + 0.466i)7-s + (−0.250 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.535 − 0.462i)10-s + (−0.0170 − 0.0295i)11-s + (0.0747 + 0.278i)12-s + (−0.717 − 0.717i)13-s + (−0.480 − 0.518i)14-s + (−0.567 + 0.108i)15-s + (0.124 − 0.216i)16-s + (−1.75 − 0.468i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.566 + 0.824i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (163, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.566 + 0.824i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.266445 - 0.506280i\)
\(L(\frac12)\)  \(\approx\)  \(0.266445 - 0.506280i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
3 \( 1 + (-0.448 + 1.67i)T \)
5 \( 1 + (2.17 + 4.50i)T \)
7 \( 1 + (6.19 - 3.26i)T \)
good11 \( 1 + (0.187 + 0.325i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (9.32 + 9.32i)T + 169iT^{2} \)
17 \( 1 + (29.7 + 7.97i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (6.46 + 3.73i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-38.4 + 10.3i)T + (458. - 264.5i)T^{2} \)
29 \( 1 - 22.0iT - 841T^{2} \)
31 \( 1 + (23.7 + 41.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-1.58 - 5.90i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 - 78.1T + 1.68e3T^{2} \)
43 \( 1 + (12.2 + 12.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (-3.06 - 11.4i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (1.05 - 3.93i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (2.87 - 1.65i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (44.2 - 76.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (86.4 + 23.1i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 64.3T + 5.04e3T^{2} \)
73 \( 1 + (-3.13 + 11.6i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-31.3 - 18.1i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (94.4 + 94.4i)T + 6.88e3iT^{2} \)
89 \( 1 + (-50.7 - 29.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-56.0 + 56.0i)T - 9.40e3iT^{2} \)
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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

−12.21195509452428273415199481869, −10.99104959747899412555815027294, −9.265335304045985185149821026985, −8.878274951445966881252667602350, −7.63166719557850871720236812467, −6.75119338217302221426349227779, −5.55679925018608747076895923224, −4.42388767098710475304778173771, −2.75217639333192537377165442324, −0.28078621828448502309068435699, 2.50609469506330517817324582466, 3.66007837387315340803482274226, 4.61686301726599909934170008235, 6.37088478235909897800963361545, 7.29379470556680394612316906695, 8.891202180487520280757298957681, 9.686179480240317535456373025444, 10.77166062344540761070479217961, 11.16112829069572413206122812844, 12.45297908738403380335776920248

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