Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.951 - 0.306i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 − 1.36i)2-s + (0.448 + 1.67i)3-s + (−1.73 − i)4-s + (4.81 + 1.32i)5-s + 2.44·6-s + (4.03 + 5.72i)7-s + (−2 + 1.99i)8-s + (−2.59 + 1.50i)9-s + (3.58 − 6.09i)10-s + (−5.44 + 9.42i)11-s + (0.896 − 3.34i)12-s + (−4.13 + 4.13i)13-s + (9.29 − 3.41i)14-s + (−0.0638 + 8.66i)15-s + (1.99 + 3.46i)16-s + (1.82 − 0.489i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.149 + 0.557i)3-s + (−0.433 − 0.250i)4-s + (0.963 + 0.265i)5-s + 0.408·6-s + (0.576 + 0.817i)7-s + (−0.250 + 0.249i)8-s + (−0.288 + 0.166i)9-s + (0.358 − 0.609i)10-s + (−0.494 + 0.856i)11-s + (0.0747 − 0.278i)12-s + (−0.318 + 0.318i)13-s + (0.663 − 0.243i)14-s + (−0.00425 + 0.577i)15-s + (0.124 + 0.216i)16-s + (0.107 − 0.0287i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.951 - 0.306i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.951 - 0.306i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.95547 + 0.306863i\)
\(L(\frac12)\)  \(\approx\)  \(1.95547 + 0.306863i\)
\(L(2)\)   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 \( 1 + (-0.366 + 1.36i)T \)
3 \( 1 + (-0.448 - 1.67i)T \)
5 \( 1 + (-4.81 - 1.32i)T \)
7 \( 1 + (-4.03 - 5.72i)T \)
good11 \( 1 + (5.44 - 9.42i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (4.13 - 4.13i)T - 169iT^{2} \)
17 \( 1 + (-1.82 + 0.489i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-27.1 + 15.6i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-21.8 - 5.85i)T + (458. + 264.5i)T^{2} \)
29 \( 1 + 35.4iT - 841T^{2} \)
31 \( 1 + (7.10 - 12.3i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (3.60 - 13.4i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 75.8T + 1.68e3T^{2} \)
43 \( 1 + (5.54 - 5.54i)T - 1.84e3iT^{2} \)
47 \( 1 + (-23.9 + 89.5i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (19.6 + 73.5i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (39.8 + 22.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-46.7 - 80.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-24.1 + 6.47i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 4.10T + 5.04e3T^{2} \)
73 \( 1 + (25.3 + 94.5i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (25.8 - 14.9i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (42.2 - 42.2i)T - 6.88e3iT^{2} \)
89 \( 1 + (71.6 - 41.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (60.0 + 60.0i)T + 9.40e3iT^{2} \)
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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

−12.00812405216534315112722599244, −11.26812118209584859736590670501, −10.09051142071345525252274863429, −9.558618907297428034846095736485, −8.580986202289075193552585270067, −7.05548578023506978346217448274, −5.42633582894608711300514975107, −4.87368579394853772304275722730, −3.03492796587960984709865855195, −1.94649717581973929963504941344, 1.19384467175426224165255249324, 3.17213924036620222350922852259, 4.98575821751517294619296286327, 5.79962577275017760586797780788, 7.05528530196027300034488796692, 7.912012929671654138684281513898, 8.902025632931180232530580140025, 10.05502203117283317601220846928, 11.08773941325309705781790719101, 12.45237614905146627686009750665

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