Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.942 - 0.334i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 + 0.366i)2-s + (1.67 + 0.448i)3-s + (1.73 − i)4-s + (−2.69 − 4.20i)5-s − 2.44·6-s + (1.39 + 6.85i)7-s + (−1.99 + 2i)8-s + (2.59 + 1.50i)9-s + (5.22 + 4.76i)10-s + (6.05 + 10.4i)11-s + (3.34 − 0.896i)12-s + (12.6 − 12.6i)13-s + (−4.42 − 8.85i)14-s + (−2.62 − 8.25i)15-s + (1.99 − 3.46i)16-s + (4.41 − 16.4i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.557 + 0.149i)3-s + (0.433 − 0.250i)4-s + (−0.539 − 0.841i)5-s − 0.408·6-s + (0.199 + 0.979i)7-s + (−0.249 + 0.250i)8-s + (0.288 + 0.166i)9-s + (0.522 + 0.476i)10-s + (0.550 + 0.953i)11-s + (0.278 − 0.0747i)12-s + (0.971 − 0.971i)13-s + (−0.315 − 0.632i)14-s + (−0.175 − 0.550i)15-s + (0.124 − 0.216i)16-s + (0.259 − 0.968i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.942 - 0.334i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.942 - 0.334i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.34804 + 0.231982i\)
\(L(\frac12)\)  \(\approx\)  \(1.34804 + 0.231982i\)
\(L(2)\)   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 \( 1 + (1.36 - 0.366i)T \)
3 \( 1 + (-1.67 - 0.448i)T \)
5 \( 1 + (2.69 + 4.20i)T \)
7 \( 1 + (-1.39 - 6.85i)T \)
good11 \( 1 + (-6.05 - 10.4i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-12.6 + 12.6i)T - 169iT^{2} \)
17 \( 1 + (-4.41 + 16.4i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (-30.0 - 17.3i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-1.08 - 4.04i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + 30.5iT - 841T^{2} \)
31 \( 1 + (-11.3 - 19.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-40.8 + 10.9i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 68.1T + 1.68e3T^{2} \)
43 \( 1 + (48.6 - 48.6i)T - 1.84e3iT^{2} \)
47 \( 1 + (58.4 - 15.6i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (38.0 + 10.1i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (50.9 - 29.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-18.7 + 32.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-29.4 + 109. i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 27.4T + 5.04e3T^{2} \)
73 \( 1 + (125. + 33.4i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-9.99 - 5.77i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (11.4 - 11.4i)T - 6.88e3iT^{2} \)
89 \( 1 + (-29.3 - 16.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (26.7 + 26.7i)T + 9.40e3iT^{2} \)
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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

−12.06192066767163965094736322925, −11.34601331190656479248739689170, −9.728670578314831818587619375579, −9.308907736125908762709304945446, −8.136045021487867670247520752936, −7.66065480202707266943838217612, −5.95902379809390998859907800600, −4.78483253959943424473734341316, −3.15687793231089816596194157926, −1.33891294561774971533957829076, 1.19357646452423207889704678801, 3.15003893449803841060775194201, 4.03243218671093407730332382941, 6.33030680376599699620627878660, 7.19201051633304999835158766984, 8.102334939449147233304112760529, 9.042185915600058683712815072187, 10.17255807832190843912199117243, 11.19922852578197987236453525142, 11.58127358565347927252682999938

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