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 − 12.4·11-s + (−2.44 − 2.44i)12-s + (3.13 − 3.13i)13-s + 3.74i·14-s + (−5.37 − 6.79i)15-s − 4·16-s + (−5.80 − 5.80i)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 − 1.12·11-s + (−0.204 − 0.204i)12-s + (0.241 − 0.241i)13-s + 0.267i·14-s + (−0.358 − 0.452i)15-s − 0.250·16-s + (−0.341 − 0.341i)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  :  $\chi_{210} (127, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.940 - 0.340i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.233664 + 1.33003i\)
\(L(\frac12)\)  \(\approx\)  \(0.233664 + 1.33003i\)
\(L(2)\)   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 \( 1 + 12.4T + 121T^{2} \)
13 \( 1 + (-3.13 + 3.13i)T - 169iT^{2} \)
17 \( 1 + (5.80 + 5.80i)T + 289iT^{2} \)
19 \( 1 - 26.5iT - 361T^{2} \)
23 \( 1 + (10.4 - 10.4i)T - 529iT^{2} \)
29 \( 1 - 14.5iT - 841T^{2} \)
31 \( 1 - 42.6T + 961T^{2} \)
37 \( 1 + (-11.9 - 11.9i)T + 1.36e3iT^{2} \)
41 \( 1 - 37.5T + 1.68e3T^{2} \)
43 \( 1 + (-24.0 + 24.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (-8.83 - 8.83i)T + 2.20e3iT^{2} \)
53 \( 1 + (1.97 - 1.97i)T - 2.80e3iT^{2} \)
59 \( 1 - 88.2iT - 3.48e3T^{2} \)
61 \( 1 - 102.T + 3.72e3T^{2} \)
67 \( 1 + (22.8 + 22.8i)T + 4.48e3iT^{2} \)
71 \( 1 + 10.7T + 5.04e3T^{2} \)
73 \( 1 + (-80.4 + 80.4i)T - 5.32e3iT^{2} \)
79 \( 1 - 138. iT - 6.24e3T^{2} \)
83 \( 1 + (96.0 - 96.0i)T - 6.88e3iT^{2} \)
89 \( 1 + 3.29iT - 7.92e3T^{2} \)
97 \( 1 + (88.5 + 88.5i)T + 9.40e3iT^{2} \)
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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

−12.53930150990362718083491856400, −11.60432787351453706019424424804, −10.69514067458704460754521630786, −9.875877217940338713877134116025, −8.309446717150341544182982181754, −7.44063405935776294592685769891, −6.20497961381797694874035803457, −5.37684876023716186632437867365, −4.01753136566828914490718200427, −2.69715732058352496712588823737, 0.67161952533483485419618285373, 2.36386048854697758808198580105, 4.30641664752861511289395646882, 5.12015398653510732720568641796, 6.29501124752707469534053042126, 7.69655047408575525969710040052, 8.697671635365358027899623936137, 9.958133741588331880489607249931, 11.02648067032715874509628690227, 11.74192605907796262559850200872

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