Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.0695 + 0.997i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 1.22i)2-s + (−1.5 − 0.866i)3-s + (−0.999 + 1.73i)4-s + (−1.93 + 1.11i)5-s − 2.44i·6-s + (−5.26 − 4.61i)7-s − 2.82·8-s + (1.5 + 2.59i)9-s + (−2.73 − 1.58i)10-s + (5.41 − 9.37i)11-s + (2.99 − 1.73i)12-s − 19.2i·13-s + (1.93 − 9.70i)14-s + 3.87·15-s + (−2.00 − 3.46i)16-s + (−8.89 − 5.13i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.5 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.387 + 0.223i)5-s − 0.408i·6-s + (−0.751 − 0.659i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−0.273 − 0.158i)10-s + (0.491 − 0.852i)11-s + (0.249 − 0.144i)12-s − 1.48i·13-s + (0.137 − 0.693i)14-s + 0.258·15-s + (−0.125 − 0.216i)16-s + (−0.523 − 0.302i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0695 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0695 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.0695 + 0.997i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.0695 + 0.997i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.464911 - 0.498432i\)
\(L(\frac12)\)  \(\approx\)  \(0.464911 - 0.498432i\)
\(L(2)\)   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 \( 1 + (-0.707 - 1.22i)T \)
3 \( 1 + (1.5 + 0.866i)T \)
5 \( 1 + (1.93 - 1.11i)T \)
7 \( 1 + (5.26 + 4.61i)T \)
good11 \( 1 + (-5.41 + 9.37i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 19.2iT - 169T^{2} \)
17 \( 1 + (8.89 + 5.13i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (18.0 - 10.4i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (10.5 + 18.2i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 19.0T + 841T^{2} \)
31 \( 1 + (34.6 + 20.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-25.1 - 43.6i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 22.7iT - 1.68e3T^{2} \)
43 \( 1 + 48.4T + 1.84e3T^{2} \)
47 \( 1 + (-57.6 + 33.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (2.47 - 4.28i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (24.4 + 14.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (60.6 - 35.0i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (9.65 - 16.7i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 49.4T + 5.04e3T^{2} \)
73 \( 1 + (115. + 66.4i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-45.0 - 78.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 101. iT - 6.88e3T^{2} \)
89 \( 1 + (-34.3 + 19.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 68.6iT - 9.40e3T^{2} \)
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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

−12.07890330369500679241645554842, −10.93236577924297394194011433775, −10.11926733414291986406689536658, −8.598179145858831628952898528008, −7.66951568254287239732025698372, −6.57212998341379170810776744962, −5.88590319473020126731816441527, −4.36552894805361874161059525994, −3.16037557574744135352548187731, −0.35182848913325975009181206190, 2.02682800163684990068561994024, 3.82800941830657239872694347131, 4.69960331798621384469237647252, 6.09353317995189421843631455855, 7.02684933582954126753875342191, 8.932358985399064500395414342669, 9.418140754052905712461750046869, 10.63351026061361897389119282870, 11.60492865684122469156772546864, 12.26643719277297459731292508065

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