Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.0650 + 0.997i$
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 + (−4.93 + 0.829i)5-s + 2.44·6-s + (1.87 + 1.87i)7-s + (2 − 2i)8-s − 2.99i·9-s + (5.76 + 4.10i)10-s − 5.74·11-s + (−2.44 − 2.44i)12-s + (15.0 − 15.0i)13-s − 3.74i·14-s + (5.02 − 7.05i)15-s − 4·16-s + (−4.78 − 4.78i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s + (−0.986 + 0.165i)5-s + 0.408·6-s + (0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.576 + 0.410i)10-s − 0.522·11-s + (−0.204 − 0.204i)12-s + (1.15 − 1.15i)13-s − 0.267i·14-s + (0.334 − 0.470i)15-s − 0.250·16-s + (−0.281 − 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0650 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0650 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.0650 + 0.997i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (127, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.0650 + 0.997i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.441135 - 0.470822i\)
\(L(\frac12)\)  \(\approx\)  \(0.441135 - 0.470822i\)
\(L(2)\)   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 \( 1 + (4.93 - 0.829i)T \)
7 \( 1 + (-1.87 - 1.87i)T \)
good11 \( 1 + 5.74T + 121T^{2} \)
13 \( 1 + (-15.0 + 15.0i)T - 169iT^{2} \)
17 \( 1 + (4.78 + 4.78i)T + 289iT^{2} \)
19 \( 1 + 17.4iT - 361T^{2} \)
23 \( 1 + (-13.2 + 13.2i)T - 529iT^{2} \)
29 \( 1 + 37.7iT - 841T^{2} \)
31 \( 1 + 27.0T + 961T^{2} \)
37 \( 1 + (-11.3 - 11.3i)T + 1.36e3iT^{2} \)
41 \( 1 + 53.0T + 1.68e3T^{2} \)
43 \( 1 + (-37.1 + 37.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (9.39 + 9.39i)T + 2.20e3iT^{2} \)
53 \( 1 + (-43.8 + 43.8i)T - 2.80e3iT^{2} \)
59 \( 1 - 62.9iT - 3.48e3T^{2} \)
61 \( 1 + 1.67T + 3.72e3T^{2} \)
67 \( 1 + (-28.4 - 28.4i)T + 4.48e3iT^{2} \)
71 \( 1 - 47.2T + 5.04e3T^{2} \)
73 \( 1 + (31.2 - 31.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 107. iT - 6.24e3T^{2} \)
83 \( 1 + (-18.2 + 18.2i)T - 6.88e3iT^{2} \)
89 \( 1 + 174. iT - 7.92e3T^{2} \)
97 \( 1 + (91.5 + 91.5i)T + 9.40e3iT^{2} \)
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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

−11.53718809086597319503572295690, −11.01227221308770591904640815131, −10.20545685997947001282759899834, −8.833415901931957212152722920183, −8.123573982670183968654440850327, −6.94725039151420426532538086699, −5.41814022951615400073468133446, −4.11453576898355198152536297033, −2.87179214096906934885447081707, −0.49606625727679552315257615614, 1.39950008238396848672532526061, 3.81590513560130877485141928478, 5.14448648271434780789362562651, 6.45781358156630169478216757405, 7.39702038495747657862796580660, 8.257839607878554625140527077275, 9.167265601581869092391595716378, 10.73171995264262040128294357648, 11.22130887202472713763661588377, 12.29365342708379248612425682045

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