Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.959 - 0.281i$
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 + (−0.258 − 4.99i)5-s − 2.44·6-s + (6.75 − 1.84i)7-s + (2 − 1.99i)8-s + (−2.59 + 1.50i)9-s + (6.91 + 1.47i)10-s + (7.12 − 12.3i)11-s + (0.896 − 3.34i)12-s + (−2.75 + 2.75i)13-s + (0.0509 + 9.89i)14-s + (8.23 − 2.67i)15-s + (1.99 + 3.46i)16-s + (23.9 − 6.41i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.149 + 0.557i)3-s + (−0.433 − 0.250i)4-s + (−0.0516 − 0.998i)5-s − 0.408·6-s + (0.964 − 0.263i)7-s + (0.250 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (0.691 + 0.147i)10-s + (0.647 − 1.12i)11-s + (0.0747 − 0.278i)12-s + (−0.212 + 0.212i)13-s + (0.00363 + 0.707i)14-s + (0.549 − 0.178i)15-s + (0.124 + 0.216i)16-s + (1.40 − 0.377i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.959 - 0.281i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.959 - 0.281i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.53878 + 0.220908i\)
\(L(\frac12)\)  \(\approx\)  \(1.53878 + 0.220908i\)
\(L(2)\)   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 \( 1 + (0.258 + 4.99i)T \)
7 \( 1 + (-6.75 + 1.84i)T \)
good11 \( 1 + (-7.12 + 12.3i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (2.75 - 2.75i)T - 169iT^{2} \)
17 \( 1 + (-23.9 + 6.41i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (0.277 - 0.159i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-13.6 - 3.64i)T + (458. + 264.5i)T^{2} \)
29 \( 1 - 24.2iT - 841T^{2} \)
31 \( 1 + (7.62 - 13.1i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-0.611 + 2.28i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 29.0T + 1.68e3T^{2} \)
43 \( 1 + (11.7 - 11.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (-21.9 + 81.7i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (19.4 + 72.6i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (31.7 + 18.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-54.2 - 93.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (66.0 - 17.6i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 20.3T + 5.04e3T^{2} \)
73 \( 1 + (15.1 + 56.5i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (44.5 - 25.7i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (82.8 - 82.8i)T - 6.88e3iT^{2} \)
89 \( 1 + (95.6 - 55.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-68.2 - 68.2i)T + 9.40e3iT^{2} \)
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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

−12.09535110661777339873160913213, −11.21051286898302517059230771526, −10.01217566885809300263862114562, −8.942352951689860962717143007998, −8.378620332070019120142186395364, −7.30339253627967713367146589013, −5.66475249701964029174707174219, −4.92542915956382847001634129372, −3.69087905487099815084254182005, −1.11062738048683047225672954823, 1.58274061973270565968604083577, 2.83280733470467703005480406416, 4.32208873953886456421540488868, 5.88474350675149039039908854972, 7.31769422339247790228056553472, 7.933439988513615827535600310847, 9.295675399590748225318978598724, 10.25312656606532870322206059458, 11.24337324858320484224249665229, 12.01274672645810992330204452538

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