Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.999 + 0.0254i$
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 + (1.65 − 4.71i)5-s + 2.44·6-s + (−6.62 + 2.26i)7-s + (2 + 1.99i)8-s + (−2.59 − 1.50i)9-s + (−7.05 − 0.538i)10-s + (−1.33 − 2.30i)11-s + (−0.896 − 3.34i)12-s + (−9.16 − 9.16i)13-s + (5.52 + 8.21i)14-s + (7.14 + 4.88i)15-s + (1.99 − 3.46i)16-s + (−2.70 − 0.724i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.149 + 0.557i)3-s + (−0.433 + 0.250i)4-s + (0.331 − 0.943i)5-s + 0.408·6-s + (−0.946 + 0.324i)7-s + (0.250 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.705 − 0.0538i)10-s + (−0.121 − 0.209i)11-s + (−0.0747 − 0.278i)12-s + (−0.704 − 0.704i)13-s + (0.394 + 0.586i)14-s + (0.476 + 0.325i)15-s + (0.124 − 0.216i)16-s + (−0.159 − 0.0426i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0254i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.999 + 0.0254i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (163, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.999 + 0.0254i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.00527955 - 0.414744i\)
\(L(\frac12)\)  \(\approx\)  \(0.00527955 - 0.414744i\)
\(L(2)\)   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 \( 1 + (-1.65 + 4.71i)T \)
7 \( 1 + (6.62 - 2.26i)T \)
good11 \( 1 + (1.33 + 2.30i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (9.16 + 9.16i)T + 169iT^{2} \)
17 \( 1 + (2.70 + 0.724i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (16.5 + 9.57i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (34.9 - 9.36i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + 12.4iT - 841T^{2} \)
31 \( 1 + (-5.33 - 9.24i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (5.88 + 21.9i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 1.17T + 1.68e3T^{2} \)
43 \( 1 + (-2.70 - 2.70i)T + 1.84e3iT^{2} \)
47 \( 1 + (-19.2 - 71.7i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (2.09 - 7.83i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-93.6 + 54.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-35.0 + 60.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (16.8 + 4.51i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 66.2T + 5.04e3T^{2} \)
73 \( 1 + (-9.37 + 34.9i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (83.6 + 48.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (83.4 + 83.4i)T + 6.88e3iT^{2} \)
89 \( 1 + (-125. - 72.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (43.6 - 43.6i)T - 9.40e3iT^{2} \)
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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

−11.75077743181462850645468831119, −10.47898787325027655702240158225, −9.745371026243705659767108366205, −9.020026768180159602091609470592, −7.980910486614028258398879738427, −6.18491154579081437121573422503, −5.12790275210769730816035520590, −3.90980521734623947332236629009, −2.42484169538977523137473663643, −0.23404371264569285086575073358, 2.28550421017714030012988194271, 4.01034215075996263453753515869, 5.76865168793090300849699588947, 6.66291324058915958547762069603, 7.22784539732525676984169322946, 8.486101433141024080297400747403, 9.841985975028072141097995343742, 10.35446422328906737043768198005, 11.74052058844367994190161873031, 12.76786267605509706081982387534

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