Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.932 + 0.360i$
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 + (1.94 − 6.72i)7-s + 2.82·8-s + (1.5 + 2.59i)9-s + (−2.73 − 1.58i)10-s + (0.263 − 0.455i)11-s + (2.99 − 1.73i)12-s − 4.22i·13-s + (−9.61 + 2.37i)14-s − 3.87·15-s + (−2.00 − 3.46i)16-s + (−28.8 − 16.6i)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 + (0.277 − 0.960i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−0.273 − 0.158i)10-s + (0.0239 − 0.0414i)11-s + (0.249 − 0.144i)12-s − 0.324i·13-s + (−0.686 + 0.169i)14-s − 0.258·15-s + (−0.125 − 0.216i)16-s + (−1.69 − 0.980i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.360i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.932 + 0.360i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.932 + 0.360i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.150766 - 0.808129i\)
\(L(\frac12)\)  \(\approx\)  \(0.150766 - 0.808129i\)
\(L(2)\)   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 \( 1 + (-1.94 + 6.72i)T \)
good11 \( 1 + (-0.263 + 0.455i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 4.22iT - 169T^{2} \)
17 \( 1 + (28.8 + 16.6i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-2.75 + 1.58i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (5.52 + 9.57i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 56.1T + 841T^{2} \)
31 \( 1 + (1.63 + 0.945i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-4.87 - 8.44i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 4.07iT - 1.68e3T^{2} \)
43 \( 1 - 46.3T + 1.84e3T^{2} \)
47 \( 1 + (-54.7 + 31.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-23.2 + 40.2i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (43.7 + 25.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-89.4 + 51.6i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (4.36 - 7.56i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 29.0T + 5.04e3T^{2} \)
73 \( 1 + (-14.4 - 8.36i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-66.1 - 114. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 12.4iT - 6.88e3T^{2} \)
89 \( 1 + (59.1 - 34.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 149. iT - 9.40e3T^{2} \)
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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

−11.41749819546440817154600024268, −10.93960411325439653233998739039, −9.883079336380291606700986789514, −8.904714735363548136600403412417, −7.64776084957867430448392345969, −6.71611285044412418057529257464, −5.21022824700824489611812784964, −4.03910137817301983865643175333, −2.16786917271214546274879760060, −0.53948927312179983093619001115, 2.04009372462701913533350307535, 4.20703997094931669407528897596, 5.55238201478687297349435820419, 6.23600928500187552949409558000, 7.44845826488492735534309589622, 8.815801408897252893199356755873, 9.380906908641913564339584779674, 10.66262930604696723156486819056, 11.37028871525925703440486603152, 12.56901941162303429208321124252

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