Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.453 - 0.891i$
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 + (2.01 + 4.57i)5-s − 2.44·6-s + (−6.19 + 3.26i)7-s + (−1.99 + 2i)8-s + (2.59 + 1.50i)9-s + (−4.43 − 5.50i)10-s + (1.81 + 3.15i)11-s + (3.34 − 0.896i)12-s + (−14.2 + 14.2i)13-s + (7.26 − 6.72i)14-s + (1.32 + 8.55i)15-s + (1.99 − 3.46i)16-s + (5.53 − 20.6i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.557 + 0.149i)3-s + (0.433 − 0.250i)4-s + (0.403 + 0.914i)5-s − 0.408·6-s + (−0.884 + 0.466i)7-s + (−0.249 + 0.250i)8-s + (0.288 + 0.166i)9-s + (−0.443 − 0.550i)10-s + (0.165 + 0.286i)11-s + (0.278 − 0.0747i)12-s + (−1.09 + 1.09i)13-s + (0.518 − 0.480i)14-s + (0.0884 + 0.570i)15-s + (0.124 − 0.216i)16-s + (0.325 − 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.453 - 0.891i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.453 - 0.891i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.570939 + 0.931544i\)
\(L(\frac12)\)  \(\approx\)  \(0.570939 + 0.931544i\)
\(L(2)\)   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 \( 1 + (-2.01 - 4.57i)T \)
7 \( 1 + (6.19 - 3.26i)T \)
good11 \( 1 + (-1.81 - 3.15i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (14.2 - 14.2i)T - 169iT^{2} \)
17 \( 1 + (-5.53 + 20.6i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (0.949 + 0.548i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-6.52 - 24.3i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + 1.99iT - 841T^{2} \)
31 \( 1 + (-25.3 - 43.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (44.3 - 11.8i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 18.6T + 1.68e3T^{2} \)
43 \( 1 + (6.49 - 6.49i)T - 1.84e3iT^{2} \)
47 \( 1 + (-1.13 + 0.305i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-91.2 - 24.4i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-95.2 + 54.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-55.8 + 96.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (3.19 - 11.9i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 69.2T + 5.04e3T^{2} \)
73 \( 1 + (-46.3 - 12.4i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (64.8 + 37.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (28.7 - 28.7i)T - 6.88e3iT^{2} \)
89 \( 1 + (24.1 + 13.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (101. + 101. i)T + 9.40e3iT^{2} \)
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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

−12.27173957410459601431085897699, −11.41608768586843697918282045905, −9.928440891083431920332445496323, −9.740789884745215096110651890719, −8.707885317153530331023194179135, −7.14205538116095982709013081960, −6.80891442503891660693369329511, −5.23278031303048188020393075050, −3.28133201487084882574399895794, −2.18982665735559844982557096810, 0.69645499207648695387682237822, 2.47700541862355326586059382809, 3.94469475039332370276659855585, 5.63982975812899924338129035650, 6.90527680794419076603287620668, 8.087357132716542870682701654488, 8.806425620883995012267176716123, 9.971533394168874572053323794644, 10.31717294683706931766673140851, 12.04260009822090648734612182700

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