Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.497 + 0.867i$
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 + (−2.84 + 0.962i)3-s + (−1.73 + i)4-s + (−4.69 − 1.71i)5-s + (−2.35 − 3.52i)6-s + (−1.00 + 6.92i)7-s + (−2 − 1.99i)8-s + (7.14 − 5.46i)9-s + (0.626 − 7.04i)10-s + (8.28 − 4.78i)11-s + (3.95 − 4.50i)12-s + (−9.08 − 9.08i)13-s + (−9.83 + 1.16i)14-s + (14.9 + 0.359i)15-s + (1.99 − 3.46i)16-s + (6.55 − 24.4i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.947 + 0.320i)3-s + (−0.433 + 0.250i)4-s + (−0.939 − 0.343i)5-s + (−0.392 − 0.588i)6-s + (−0.143 + 0.989i)7-s + (−0.250 − 0.249i)8-s + (0.794 − 0.607i)9-s + (0.0626 − 0.704i)10-s + (0.753 − 0.434i)11-s + (0.329 − 0.375i)12-s + (−0.698 − 0.698i)13-s + (−0.702 + 0.0832i)14-s + (0.999 + 0.0239i)15-s + (0.124 − 0.216i)16-s + (0.385 − 1.44i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.497 + 0.867i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (47, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.497 + 0.867i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.402685 - 0.233246i\)
\(L(\frac12)\)  \(\approx\)  \(0.402685 - 0.233246i\)
\(L(2)\)   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 \( 1 + (2.84 - 0.962i)T \)
5 \( 1 + (4.69 + 1.71i)T \)
7 \( 1 + (1.00 - 6.92i)T \)
good11 \( 1 + (-8.28 + 4.78i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (9.08 + 9.08i)T + 169iT^{2} \)
17 \( 1 + (-6.55 + 24.4i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (-10.4 + 18.1i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (33.1 - 8.88i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + 6.08T + 841T^{2} \)
31 \( 1 + (-7.22 + 4.17i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (14.2 - 3.80i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 55.3T + 1.68e3T^{2} \)
43 \( 1 + (-57.2 + 57.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (74.4 - 19.9i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (0.192 - 0.717i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-88.7 + 51.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (92.7 + 53.5i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-6.19 + 23.1i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 26.3iT - 5.04e3T^{2} \)
73 \( 1 + (2.02 - 7.55i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (34.5 + 19.9i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (52.9 - 52.9i)T - 6.88e3iT^{2} \)
89 \( 1 + (-24.2 - 14.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (19.4 - 19.4i)T - 9.40e3iT^{2} \)
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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

−11.90615041144553730612249648736, −11.45226249295778595292527171361, −9.836312779564629437560846951268, −9.013423600454613937920192774164, −7.78146330102849296360391880754, −6.76140837703708045129273627636, −5.53162302486504494198118437356, −4.82018205789561335711895756471, −3.39940986308320079192534687326, −0.29309916439646377069518473889, 1.52277830764417503504501348312, 3.78779384160033925877558731092, 4.47903862484469651988244922194, 6.14178932711898364538339157235, 7.15043483993740677827023834288, 8.114604463715267000443443150991, 9.934742260529547358629557747081, 10.43701514181553171252267382976, 11.57501789451834970401187403567, 12.07179042633468109937345495892

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