Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.461 + 0.887i$
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 + (−0.448 + 1.67i)3-s + (−1.73 + i)4-s + (−1.45 − 4.78i)5-s − 2.44·6-s + (−1.05 − 6.92i)7-s + (−2 − 1.99i)8-s + (−2.59 − 1.50i)9-s + (6.00 − 3.74i)10-s + (−6.50 − 11.2i)11-s + (−0.896 − 3.34i)12-s + (−0.863 − 0.863i)13-s + (9.06 − 3.96i)14-s + (8.65 − 0.293i)15-s + (1.99 − 3.46i)16-s + (−0.902 − 0.241i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.149 + 0.557i)3-s + (−0.433 + 0.250i)4-s + (−0.291 − 0.956i)5-s − 0.408·6-s + (−0.150 − 0.988i)7-s + (−0.250 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.600 − 0.374i)10-s + (−0.591 − 1.02i)11-s + (−0.0747 − 0.278i)12-s + (−0.0664 − 0.0664i)13-s + (0.647 − 0.283i)14-s + (0.577 − 0.0195i)15-s + (0.124 − 0.216i)16-s + (−0.0531 − 0.0142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 + 0.887i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.461 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.461 + 0.887i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (163, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.461 + 0.887i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.781204 - 0.474038i\)
\(L(\frac12)\)  \(\approx\)  \(0.781204 - 0.474038i\)
\(L(2)\)   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 \( 1 + (0.448 - 1.67i)T \)
5 \( 1 + (1.45 + 4.78i)T \)
7 \( 1 + (1.05 + 6.92i)T \)
good11 \( 1 + (6.50 + 11.2i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (0.863 + 0.863i)T + 169iT^{2} \)
17 \( 1 + (0.902 + 0.241i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (5.84 + 3.37i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-14.5 + 3.90i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + 31.3iT - 841T^{2} \)
31 \( 1 + (-23.4 - 40.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (9.80 + 36.5i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 64.3T + 1.68e3T^{2} \)
43 \( 1 + (-17.7 - 17.7i)T + 1.84e3iT^{2} \)
47 \( 1 + (12.9 + 48.1i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-19.3 + 72.2i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-31.0 + 17.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (54.5 - 94.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-60.4 - 16.2i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 74.7T + 5.04e3T^{2} \)
73 \( 1 + (-21.5 + 80.2i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-83.9 - 48.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-31.9 - 31.9i)T + 6.88e3iT^{2} \)
89 \( 1 + (44.1 + 25.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (83.5 - 83.5i)T - 9.40e3iT^{2} \)
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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

−12.05504779947169348894652518273, −10.93062553085781407970460299370, −9.970411450330412084873555347535, −8.773269181314593059810542052256, −8.068383307758744043605959477404, −6.81208131684245247828826958560, −5.49867747495015822115799693116, −4.58690471437317897323472815958, −3.48093690270757112833405729001, −0.48328304602846934425516382099, 2.07876536354279221135583665389, 3.11231256924189122392530957388, 4.81545401523669008837137480746, 6.10370883910866855039974249962, 7.18118349194200621754172888844, 8.315222824015919000390183926887, 9.587078307373438382920767358598, 10.53068574308168995037508838557, 11.49633239968153043998244071272, 12.24235533946003917778887378887

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