Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.596 + 0.802i$
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 + (0.947 − 2.84i)3-s − 2i·4-s + (4.64 − 1.84i)5-s + (−1.89 − 3.79i)6-s + (−6.25 − 3.13i)7-s + (−2 − 2i)8-s + (−7.20 − 5.39i)9-s + (2.79 − 6.49i)10-s + 2.08i·11-s + (−5.69 − 1.89i)12-s + (8.39 + 8.39i)13-s + (−9.39 + 3.12i)14-s + (−0.854 − 14.9i)15-s − 4·16-s + (4.96 + 4.96i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.315 − 0.948i)3-s − 0.5i·4-s + (0.929 − 0.369i)5-s + (−0.316 − 0.632i)6-s + (−0.893 − 0.448i)7-s + (−0.250 − 0.250i)8-s + (−0.800 − 0.599i)9-s + (0.279 − 0.649i)10-s + 0.189i·11-s + (−0.474 − 0.157i)12-s + (0.645 + 0.645i)13-s + (−0.671 + 0.222i)14-s + (−0.0569 − 0.998i)15-s − 0.250·16-s + (0.292 + 0.292i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.596 + 0.802i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (167, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.596 + 0.802i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.00649 - 2.00303i\)
\(L(\frac12)\)  \(\approx\)  \(1.00649 - 2.00303i\)
\(L(2)\)   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 \( 1 + (-0.947 + 2.84i)T \)
5 \( 1 + (-4.64 + 1.84i)T \)
7 \( 1 + (6.25 + 3.13i)T \)
good11 \( 1 - 2.08iT - 121T^{2} \)
13 \( 1 + (-8.39 - 8.39i)T + 169iT^{2} \)
17 \( 1 + (-4.96 - 4.96i)T + 289iT^{2} \)
19 \( 1 - 17.3T + 361T^{2} \)
23 \( 1 + (3.08 + 3.08i)T + 529iT^{2} \)
29 \( 1 + 39.1T + 841T^{2} \)
31 \( 1 + 42.3iT - 961T^{2} \)
37 \( 1 + (-36.7 - 36.7i)T + 1.36e3iT^{2} \)
41 \( 1 - 15.5T + 1.68e3T^{2} \)
43 \( 1 + (-22.8 + 22.8i)T - 1.84e3iT^{2} \)
47 \( 1 + (-33.4 - 33.4i)T + 2.20e3iT^{2} \)
53 \( 1 + (-59.7 - 59.7i)T + 2.80e3iT^{2} \)
59 \( 1 - 48.9iT - 3.48e3T^{2} \)
61 \( 1 + 82.9iT - 3.72e3T^{2} \)
67 \( 1 + (54.8 + 54.8i)T + 4.48e3iT^{2} \)
71 \( 1 + 74.9iT - 5.04e3T^{2} \)
73 \( 1 + (-75.1 - 75.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 3.61iT - 6.24e3T^{2} \)
83 \( 1 + (103. - 103. i)T - 6.88e3iT^{2} \)
89 \( 1 - 24.4iT - 7.92e3T^{2} \)
97 \( 1 + (35.3 - 35.3i)T - 9.40e3iT^{2} \)
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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

−12.10347896070416055180365311420, −10.98626846383113899842410369322, −9.702491519458249906797937525965, −9.123148891079779403150688834967, −7.60900854797292854050128769744, −6.41437892383259325885980938675, −5.70342216697372104691479374792, −3.92281298751125272435384776460, −2.52378713945907049046085331946, −1.14133172483907125353069157791, 2.74404710966424375294900753714, 3.66921823500557824232008791983, 5.40691795744999710021241460638, 5.89340020542541934027771848918, 7.29174369560188543347751799549, 8.721330970635145102384605772951, 9.505178964676561283243015003891, 10.36698840966710295365496932006, 11.43999544138477039365760872428, 12.81375682505648843907296532935

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