Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.739 + 0.673i$
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 + (−2.35 − 1.86i)3-s − 2i·4-s + (−1.91 + 4.61i)5-s + (4.21 − 0.488i)6-s + (6.26 − 3.12i)7-s + (2 + 2i)8-s + (2.05 + 8.76i)9-s + (−2.70 − 6.53i)10-s + 0.117i·11-s + (−3.72 + 4.70i)12-s + (−9.72 − 9.72i)13-s + (−3.13 + 9.39i)14-s + (13.1 − 7.29i)15-s − 4·16-s + (−13.8 − 13.8i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.783 − 0.620i)3-s − 0.5i·4-s + (−0.383 + 0.923i)5-s + (0.702 − 0.0814i)6-s + (0.894 − 0.446i)7-s + (0.250 + 0.250i)8-s + (0.228 + 0.973i)9-s + (−0.270 − 0.653i)10-s + 0.0106i·11-s + (−0.310 + 0.391i)12-s + (−0.747 − 0.747i)13-s + (−0.223 + 0.670i)14-s + (0.873 − 0.486i)15-s − 0.250·16-s + (−0.815 − 0.815i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.739 + 0.673i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (167, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.739 + 0.673i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0746974 - 0.192907i\)
\(L(\frac12)\)  \(\approx\)  \(0.0746974 - 0.192907i\)
\(L(2)\)   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 \( 1 + (2.35 + 1.86i)T \)
5 \( 1 + (1.91 - 4.61i)T \)
7 \( 1 + (-6.26 + 3.12i)T \)
good11 \( 1 - 0.117iT - 121T^{2} \)
13 \( 1 + (9.72 + 9.72i)T + 169iT^{2} \)
17 \( 1 + (13.8 + 13.8i)T + 289iT^{2} \)
19 \( 1 + 29.7T + 361T^{2} \)
23 \( 1 + (-2.25 - 2.25i)T + 529iT^{2} \)
29 \( 1 + 46.0T + 841T^{2} \)
31 \( 1 - 1.50iT - 961T^{2} \)
37 \( 1 + (-5.32 - 5.32i)T + 1.36e3iT^{2} \)
41 \( 1 - 13.4T + 1.68e3T^{2} \)
43 \( 1 + (-36.8 + 36.8i)T - 1.84e3iT^{2} \)
47 \( 1 + (29.7 + 29.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (59.8 + 59.8i)T + 2.80e3iT^{2} \)
59 \( 1 - 84.9iT - 3.48e3T^{2} \)
61 \( 1 - 34.8iT - 3.72e3T^{2} \)
67 \( 1 + (34.4 + 34.4i)T + 4.48e3iT^{2} \)
71 \( 1 - 77.6iT - 5.04e3T^{2} \)
73 \( 1 + (-41.3 - 41.3i)T + 5.32e3iT^{2} \)
79 \( 1 + 0.865iT - 6.24e3T^{2} \)
83 \( 1 + (99.0 - 99.0i)T - 6.88e3iT^{2} \)
89 \( 1 + 129. iT - 7.92e3T^{2} \)
97 \( 1 + (15.7 - 15.7i)T - 9.40e3iT^{2} \)
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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

−11.39767648787919647233677737270, −10.95193047210870313123011777415, −10.05312069459990700724798014421, −8.428053680153044178112313061911, −7.44999596233507714443575898702, −6.93459433394496721619305764793, −5.68935547842031465906082344268, −4.45868180333766214539810120355, −2.18574760456272054094297227575, −0.14468719130353849226339487676, 1.82852286714088124826720478337, 4.13887324122572993106645291043, 4.79939856611963056535882142135, 6.20108371416085314070798713945, 7.76278121478456927819418519694, 8.848064100788085798047780661613, 9.460909312485527811644730629081, 10.88140563689324750702252491283, 11.31026259820112416891729285050, 12.35095329793307739362624881490

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