Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.966 - 0.255i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 1.22i)2-s + (−1.5 − 0.866i)3-s + (−0.999 + 1.73i)4-s + (1.93 − 1.11i)5-s + 2.44i·6-s + (−2.67 + 6.46i)7-s + 2.82·8-s + (1.5 + 2.59i)9-s + (−2.73 − 1.58i)10-s + (−0.578 + 1.00i)11-s + (2.99 − 1.73i)12-s + 14.8i·13-s + (9.81 − 1.29i)14-s − 3.87·15-s + (−2.00 − 3.46i)16-s + (10.9 + 6.30i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.5 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + 0.408i·6-s + (−0.381 + 0.924i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−0.273 − 0.158i)10-s + (−0.0525 + 0.0910i)11-s + (0.249 − 0.144i)12-s + 1.13i·13-s + (0.700 − 0.0928i)14-s − 0.258·15-s + (−0.125 − 0.216i)16-s + (0.642 + 0.371i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.255i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.966 - 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.966 - 0.255i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.966 - 0.255i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.00011 + 0.129955i\)
\(L(\frac12)\)  \(\approx\)  \(1.00011 + 0.129955i\)
\(L(2)\)   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 \( 1 + (0.707 + 1.22i)T \)
3 \( 1 + (1.5 + 0.866i)T \)
5 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 + (2.67 - 6.46i)T \)
good11 \( 1 + (0.578 - 1.00i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 14.8iT - 169T^{2} \)
17 \( 1 + (-10.9 - 6.30i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-16.7 + 9.65i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-12.1 - 21.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 49.0T + 841T^{2} \)
31 \( 1 + (24.9 + 14.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-26.6 - 46.1i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 38.0iT - 1.68e3T^{2} \)
43 \( 1 + 63.5T + 1.84e3T^{2} \)
47 \( 1 + (-21.8 + 12.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (10.4 - 18.0i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-21.1 - 12.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-5.53 + 3.19i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (62.2 - 107. i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 118.T + 5.04e3T^{2} \)
73 \( 1 + (34.2 + 19.7i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-46.4 - 80.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 5.79iT - 6.88e3T^{2} \)
89 \( 1 + (-131. + 75.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 144. iT - 9.40e3T^{2} \)
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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

−11.88435002968969051753554055903, −11.55413511305155380875276336071, −10.11611562771745821536840676257, −9.397917515300997108861673595258, −8.432204049660484499426491751951, −7.06560452724153574232013726153, −5.91659608005360158205937731849, −4.75240676997816358159554769429, −2.96349713982430887246875798612, −1.46185547647178607828312924954, 0.75788115018570857711974294500, 3.31164782553778583425388238616, 4.90715262175767980128875569210, 5.92895256061375172567518758168, 6.98059723813893459121939328439, 7.909561886582312174174054927721, 9.267232701569307919465917503875, 10.31110022581187112871871041201, 10.62389016755452554468300211074, 12.12033975045347638389135398477

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