Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.940 - 0.340i$
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 + (0.578 + 4.96i)5-s + 2.44·6-s + (−1.87 + 1.87i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (5.54 + 4.38i)10-s + 19.5·11-s + (2.44 − 2.44i)12-s + (8.03 + 8.03i)13-s + 3.74i·14-s + (−5.37 + 6.79i)15-s − 4·16-s + (−2.19 + 2.19i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + (0.115 + 0.993i)5-s + 0.408·6-s + (−0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (0.554 + 0.438i)10-s + 1.77·11-s + (0.204 − 0.204i)12-s + (0.617 + 0.617i)13-s + 0.267i·14-s + (−0.358 + 0.452i)15-s − 0.250·16-s + (−0.129 + 0.129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.340i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.940 - 0.340i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (43, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.940 - 0.340i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.31029 + 0.405878i\)
\(L(\frac12)\)  \(\approx\)  \(2.31029 + 0.405878i\)
\(L(2)\)   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 \( 1 + (-0.578 - 4.96i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good11 \( 1 - 19.5T + 121T^{2} \)
13 \( 1 + (-8.03 - 8.03i)T + 169iT^{2} \)
17 \( 1 + (2.19 - 2.19i)T - 289iT^{2} \)
19 \( 1 - 8.25iT - 361T^{2} \)
23 \( 1 + (17.9 + 17.9i)T + 529iT^{2} \)
29 \( 1 + 19.7iT - 841T^{2} \)
31 \( 1 - 30.0T + 961T^{2} \)
37 \( 1 + (-37.2 + 37.2i)T - 1.36e3iT^{2} \)
41 \( 1 + 80.8T + 1.68e3T^{2} \)
43 \( 1 + (13.6 + 13.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (8.17 - 8.17i)T - 2.20e3iT^{2} \)
53 \( 1 + (38.8 + 38.8i)T + 2.80e3iT^{2} \)
59 \( 1 + 74.3iT - 3.48e3T^{2} \)
61 \( 1 - 97.8T + 3.72e3T^{2} \)
67 \( 1 + (67.1 - 67.1i)T - 4.48e3iT^{2} \)
71 \( 1 + 13.3T + 5.04e3T^{2} \)
73 \( 1 + (48.2 + 48.2i)T + 5.32e3iT^{2} \)
79 \( 1 - 40.2iT - 6.24e3T^{2} \)
83 \( 1 + (-34.4 - 34.4i)T + 6.88e3iT^{2} \)
89 \( 1 + 157. iT - 7.92e3T^{2} \)
97 \( 1 + (73.2 - 73.2i)T - 9.40e3iT^{2} \)
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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

−11.93216249400547279021385790158, −11.39939556639010205586175158112, −10.23152131503036682459738385113, −9.510154814323968787565926525110, −8.423840802330755137620479920821, −6.73435492947436244913914947506, −6.08499204668347701034793512382, −4.24678752068275589752388544941, −3.44562576555606804460360603355, −1.98634681396801574477360899676, 1.30810384807650183276600907175, 3.43449695687363813033309195095, 4.53720965940148724811634649469, 5.95989266738741083487321475485, 6.83790128029680957912754095308, 8.117233713559677972262168561728, 8.892053659110617992929470282790, 9.836922339080145240248891002331, 11.58250416167987071241675224127, 12.21222447466871517045853369961

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