Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.999 - 0.00411i$
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 + (4.60 − 1.95i)5-s + 2.44·6-s + (−2.49 + 6.54i)7-s + (2 − 1.99i)8-s + (−2.59 + 1.50i)9-s + (0.985 + 7.00i)10-s + (1.59 − 2.75i)11-s + (−0.896 + 3.34i)12-s + (16.7 − 16.7i)13-s + (−8.02 − 5.79i)14-s + (−5.33 − 6.82i)15-s + (1.99 + 3.46i)16-s + (14.5 − 3.89i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.149 − 0.557i)3-s + (−0.433 − 0.250i)4-s + (0.920 − 0.390i)5-s + 0.408·6-s + (−0.356 + 0.934i)7-s + (0.250 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (0.0985 + 0.700i)10-s + (0.144 − 0.250i)11-s + (−0.0747 + 0.278i)12-s + (1.28 − 1.28i)13-s + (−0.573 − 0.414i)14-s + (−0.355 − 0.454i)15-s + (0.124 + 0.216i)16-s + (0.855 − 0.229i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00411i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.00411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.999 - 0.00411i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.999 - 0.00411i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.50788 + 0.00310253i\)
\(L(\frac12)\)  \(\approx\)  \(1.50788 + 0.00310253i\)
\(L(2)\)   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 \( 1 + (-4.60 + 1.95i)T \)
7 \( 1 + (2.49 - 6.54i)T \)
good11 \( 1 + (-1.59 + 2.75i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-16.7 + 16.7i)T - 169iT^{2} \)
17 \( 1 + (-14.5 + 3.89i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-13.5 + 7.79i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-5.41 - 1.45i)T + (458. + 264.5i)T^{2} \)
29 \( 1 - 47.4iT - 841T^{2} \)
31 \( 1 + (-0.731 + 1.26i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-14.3 + 53.3i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 27.2T + 1.68e3T^{2} \)
43 \( 1 + (16.8 - 16.8i)T - 1.84e3iT^{2} \)
47 \( 1 + (7.04 - 26.2i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (8.92 + 33.3i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-46.3 - 26.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-20.4 - 35.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-59.1 + 15.8i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 102.T + 5.04e3T^{2} \)
73 \( 1 + (7.92 + 29.5i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (101. - 58.3i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (74.9 - 74.9i)T - 6.88e3iT^{2} \)
89 \( 1 + (91.7 - 53.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (66.3 + 66.3i)T + 9.40e3iT^{2} \)
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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

−12.50873738559104586791054681207, −11.14666348646275332714614391950, −9.960555536984750472200220638384, −8.961170763225052391823577737709, −8.252319001677237734529438320499, −6.90536126819288271616475800071, −5.74350253361419725773779184166, −5.41120021399926172709430665947, −3.07960329783783520422468180573, −1.15314953742729619255252851704, 1.45379186528364725593391675206, 3.28052050858269486008194478015, 4.32507843453356389984083910442, 5.86693560900204703645893985506, 6.92514009228232698050266870886, 8.444393849913060412729218574653, 9.727139084172103137112994741191, 9.996201108891073387624340092968, 11.07375275985487566641741749750, 11.84598240302015362196186801874

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