Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.688 + 0.725i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)2-s + (1.22 + 1.22i)3-s − 2i·4-s + (4.32 − 2.51i)5-s + 2.44·6-s + (1.87 − 1.87i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (1.80 − 6.83i)10-s + 2.92·11-s + (2.44 − 2.44i)12-s + (−1.13 − 1.13i)13-s − 3.74i·14-s + (8.37 + 2.20i)15-s − 4·16-s + (1.54 − 1.54i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + (0.864 − 0.503i)5-s + 0.408·6-s + (0.267 − 0.267i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (0.180 − 0.683i)10-s + 0.265·11-s + (0.204 − 0.204i)12-s + (−0.0871 − 0.0871i)13-s − 0.267i·14-s + (0.558 + 0.147i)15-s − 0.250·16-s + (0.0908 − 0.0908i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.688 + 0.725i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (43, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.688 + 0.725i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.36764 - 1.01705i\)
\(L(\frac12)\)  \(\approx\)  \(2.36764 - 1.01705i\)
\(L(2)\)   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 \( 1 + (-1 + i)T \)
3 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 + (-4.32 + 2.51i)T \)
7 \( 1 + (-1.87 + 1.87i)T \)
good11 \( 1 - 2.92T + 121T^{2} \)
13 \( 1 + (1.13 + 1.13i)T + 169iT^{2} \)
17 \( 1 + (-1.54 + 1.54i)T - 289iT^{2} \)
19 \( 1 + 3.35iT - 361T^{2} \)
23 \( 1 + (-7.90 - 7.90i)T + 529iT^{2} \)
29 \( 1 - 13.5iT - 841T^{2} \)
31 \( 1 - 15.7T + 961T^{2} \)
37 \( 1 + (16.8 - 16.8i)T - 1.36e3iT^{2} \)
41 \( 1 + 72.6T + 1.68e3T^{2} \)
43 \( 1 + (-20.3 - 20.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (52.3 - 52.3i)T - 2.20e3iT^{2} \)
53 \( 1 + (40.5 + 40.5i)T + 2.80e3iT^{2} \)
59 \( 1 - 117. iT - 3.48e3T^{2} \)
61 \( 1 + 45.4T + 3.72e3T^{2} \)
67 \( 1 + (-57.7 + 57.7i)T - 4.48e3iT^{2} \)
71 \( 1 + 51.7T + 5.04e3T^{2} \)
73 \( 1 + (-72.3 - 72.3i)T + 5.32e3iT^{2} \)
79 \( 1 - 37.1iT - 6.24e3T^{2} \)
83 \( 1 + (21.3 + 21.3i)T + 6.88e3iT^{2} \)
89 \( 1 - 100. iT - 7.92e3T^{2} \)
97 \( 1 + (-52.5 + 52.5i)T - 9.40e3iT^{2} \)
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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

−12.10308878917076866292477555378, −10.99840000404401805780312725396, −10.04900173106769210218662660679, −9.303289986839872025524254967258, −8.251485264899906117148053027422, −6.69488137474184101702921364504, −5.38117680416497021484803468567, −4.49094011885412783433562128268, −3.04183722672410897991439383008, −1.52241950878432744418559095692, 2.00239829246164356704608878141, 3.36065343991940842874867230240, 5.00673082250196025744122644956, 6.17952208867212590912499984636, 6.97739080529161034233833321995, 8.155386301712536607046842901891, 9.159300311934763592370034973994, 10.25176153845689822202342507920, 11.49609745462188672319473877207, 12.49497129388330151931224139168

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