Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.103 - 0.994i$
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 + (−5.10 + 4.79i)7-s + 2.82·8-s + (1.5 − 2.59i)9-s + (2.73 − 1.58i)10-s + (9.03 + 15.6i)11-s + (−2.99 − 1.73i)12-s + 18.6i·13-s + (−2.26 − 9.63i)14-s − 3.87·15-s + (−2.00 + 3.46i)16-s + (1.33 − 0.770i)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 + (−0.728 + 0.684i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.273 − 0.158i)10-s + (0.821 + 1.42i)11-s + (−0.249 − 0.144i)12-s + 1.43i·13-s + (−0.161 − 0.688i)14-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (0.0785 − 0.0453i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.103 - 0.994i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (61, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.103 - 0.994i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.805635 + 0.893849i\)
\(L(\frac12)\)  \(\approx\)  \(0.805635 + 0.893849i\)
\(L(2)\)   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 \( 1 + (5.10 - 4.79i)T \)
good11 \( 1 + (-9.03 - 15.6i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 18.6iT - 169T^{2} \)
17 \( 1 + (-1.33 + 0.770i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-29.4 - 17.0i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-13.4 + 23.3i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 16.4T + 841T^{2} \)
31 \( 1 + (24.1 - 13.9i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (25.8 - 44.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 37.4iT - 1.68e3T^{2} \)
43 \( 1 + 63.6T + 1.84e3T^{2} \)
47 \( 1 + (24.4 + 14.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (0.221 + 0.384i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-64.2 + 37.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-91.0 - 52.5i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-6.35 - 11.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 45.7T + 5.04e3T^{2} \)
73 \( 1 + (-31.5 + 18.2i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-66.5 + 115. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 49.9iT - 6.88e3T^{2} \)
89 \( 1 + (-85.9 - 49.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 150. iT - 9.40e3T^{2} \)
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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

−12.28542582304778810233173038227, −11.77446551467480753268839489634, −9.946440428790948627945253306174, −9.318362626033817609497421224949, −8.516083175740743494438086706270, −7.16975102636803923418145176724, −6.65399756379478126407007079962, −5.07912342495620898077409819724, −3.69053364202186744405003327083, −1.77320143551154581594278433126, 0.76394314042301869912600220836, 3.21869694681194812059058327139, 3.56764815892428971986028067582, 5.44563992603975264443589026856, 7.06049748280871767086239308290, 7.998101170307428095117086148291, 9.112313751483194866244087151400, 9.880397529749825054270487899866, 10.95510411568660265876013782621, 11.56179248743520281697374844933

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