Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.171 + 0.985i$
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 + (−4.60 + 1.94i)5-s + 2.44·6-s + (−4.28 + 5.53i)7-s + (−1.99 − 2i)8-s + (2.59 − 1.50i)9-s + (7.00 − 0.977i)10-s + (8.47 − 14.6i)11-s + (−3.34 − 0.896i)12-s + (3.05 + 3.05i)13-s + (7.87 − 5.99i)14-s + (6.82 − 5.32i)15-s + (1.99 + 3.46i)16-s + (−2.87 − 10.7i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.557 + 0.149i)3-s + (0.433 + 0.250i)4-s + (−0.920 + 0.389i)5-s + 0.408·6-s + (−0.612 + 0.790i)7-s + (−0.249 − 0.250i)8-s + (0.288 − 0.166i)9-s + (0.700 − 0.0977i)10-s + (0.770 − 1.33i)11-s + (−0.278 − 0.0747i)12-s + (0.234 + 0.234i)13-s + (0.562 − 0.428i)14-s + (0.455 − 0.355i)15-s + (0.124 + 0.216i)16-s + (−0.169 − 0.632i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.171 + 0.985i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (193, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.171 + 0.985i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.386426 - 0.325119i\)
\(L(\frac12)\)  \(\approx\)  \(0.386426 - 0.325119i\)
\(L(2)\)   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 \( 1 + (4.60 - 1.94i)T \)
7 \( 1 + (4.28 - 5.53i)T \)
good11 \( 1 + (-8.47 + 14.6i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-3.05 - 3.05i)T + 169iT^{2} \)
17 \( 1 + (2.87 + 10.7i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (5.74 - 3.31i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-4.49 + 16.7i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + 35.8iT - 841T^{2} \)
31 \( 1 + (-22.7 + 39.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-17.7 - 4.74i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 42.9T + 1.68e3T^{2} \)
43 \( 1 + (1.98 + 1.98i)T + 1.84e3iT^{2} \)
47 \( 1 + (-42.2 - 11.3i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (82.1 - 22.0i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-22.0 - 12.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-42.4 - 73.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (16.2 + 60.8i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 137.T + 5.04e3T^{2} \)
73 \( 1 + (7.00 - 1.87i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (35.4 - 20.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (109. + 109. i)T + 6.88e3iT^{2} \)
89 \( 1 + (49.9 - 28.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (39.7 - 39.7i)T - 9.40e3iT^{2} \)
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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

−11.57812156873004337211664000126, −11.22433574622209191487962572220, −9.984072571713415876889595795038, −8.950495552797544071291684116823, −8.101275603099385207149925226635, −6.70636716491282097580712299729, −5.98597413302584758905684148006, −4.12618504024208247826211118404, −2.85005918671184478303077932009, −0.43194879121368124789647580515, 1.26771308127134281766494682040, 3.70015233691570378526728383182, 4.92159944027606600511477520658, 6.62798585422173506696089070632, 7.16869514331065257008272675354, 8.307540979487377617221685857449, 9.451091383585568653623920015863, 10.40694221510949261750744482152, 11.29059976940233993074766381698, 12.32558010359941202736472336253

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