Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.546 - 0.837i$
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 + (1.21 − 2.74i)3-s + 2i·4-s + (−3.32 + 3.73i)5-s + (−3.95 + 1.52i)6-s + (2.29 + 6.61i)7-s + (2 − 2i)8-s + (−6.03 − 6.67i)9-s + (7.05 − 0.417i)10-s + 10.5i·11-s + (5.48 + 2.43i)12-s + (−14.9 + 14.9i)13-s + (4.31 − 8.90i)14-s + (6.20 + 13.6i)15-s − 4·16-s + (−15.4 + 15.4i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.405 − 0.913i)3-s + 0.5i·4-s + (−0.664 + 0.747i)5-s + (−0.659 + 0.254i)6-s + (0.327 + 0.944i)7-s + (0.250 − 0.250i)8-s + (−0.670 − 0.741i)9-s + (0.705 − 0.0417i)10-s + 0.961i·11-s + (0.456 + 0.202i)12-s + (−1.15 + 1.15i)13-s + (0.308 − 0.636i)14-s + (0.413 + 0.910i)15-s − 0.250·16-s + (−0.911 + 0.911i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.546 - 0.837i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.546 - 0.837i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.747247 + 0.404587i\)
\(L(\frac12)\)  \(\approx\)  \(0.747247 + 0.404587i\)
\(L(2)\)   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 \( 1 + (-1.21 + 2.74i)T \)
5 \( 1 + (3.32 - 3.73i)T \)
7 \( 1 + (-2.29 - 6.61i)T \)
good11 \( 1 - 10.5iT - 121T^{2} \)
13 \( 1 + (14.9 - 14.9i)T - 169iT^{2} \)
17 \( 1 + (15.4 - 15.4i)T - 289iT^{2} \)
19 \( 1 - 17.3T + 361T^{2} \)
23 \( 1 + (-23.1 + 23.1i)T - 529iT^{2} \)
29 \( 1 - 23.7T + 841T^{2} \)
31 \( 1 - 33.1iT - 961T^{2} \)
37 \( 1 + (17.6 - 17.6i)T - 1.36e3iT^{2} \)
41 \( 1 - 11.8T + 1.68e3T^{2} \)
43 \( 1 + (22.8 + 22.8i)T + 1.84e3iT^{2} \)
47 \( 1 + (12.6 - 12.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (-15.3 + 15.3i)T - 2.80e3iT^{2} \)
59 \( 1 - 31.0iT - 3.48e3T^{2} \)
61 \( 1 + 48.6iT - 3.72e3T^{2} \)
67 \( 1 + (77.5 - 77.5i)T - 4.48e3iT^{2} \)
71 \( 1 - 60.7iT - 5.04e3T^{2} \)
73 \( 1 + (3.52 - 3.52i)T - 5.32e3iT^{2} \)
79 \( 1 + 99.4iT - 6.24e3T^{2} \)
83 \( 1 + (16.9 + 16.9i)T + 6.88e3iT^{2} \)
89 \( 1 + 17.6iT - 7.92e3T^{2} \)
97 \( 1 + (-34.7 - 34.7i)T + 9.40e3iT^{2} \)
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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

−12.04094804161693591115880244827, −11.67098452977508874564744521637, −10.39380082105130607854731031084, −9.153174233737502423650606918523, −8.391038713140010520358304527131, −7.24275601180491689261320714866, −6.64462955930385482302279485468, −4.62357986476212954630572694775, −2.90720635495984028895205840036, −1.93909392966123302319340299021, 0.52727699063809780199919785750, 3.20154590149847281091439447153, 4.63525149801214074037325148479, 5.38102256299002992945198024002, 7.35082889598979435672896920447, 7.980328898398255596717548107228, 9.005443173074745279714135369696, 9.827745979355338948380714731714, 10.89758894075004779983164520037, 11.62977519552648850496702945387

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