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  + (0.366 − 1.36i)2-s + (−0.448 − 1.67i)3-s + (−1.73 − i)4-s + (0.626 − 4.96i)5-s − 2.44·6-s + (−6.84 − 1.45i)7-s + (−2 + 1.99i)8-s + (−2.59 + 1.50i)9-s + (−6.54 − 2.67i)10-s + (−6.59 + 11.4i)11-s + (−0.896 + 3.34i)12-s + (10.7 − 10.7i)13-s + (−4.49 + 8.81i)14-s + (−8.58 + 1.17i)15-s + (1.99 + 3.46i)16-s + (−7.85 + 2.10i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.149 − 0.557i)3-s + (−0.433 − 0.250i)4-s + (0.125 − 0.992i)5-s − 0.408·6-s + (−0.978 − 0.208i)7-s + (−0.250 + 0.249i)8-s + (−0.288 + 0.166i)9-s + (−0.654 − 0.267i)10-s + (−0.599 + 1.03i)11-s + (−0.0747 + 0.278i)12-s + (0.828 − 0.828i)13-s + (−0.321 + 0.629i)14-s + (−0.572 + 0.0783i)15-s + (0.124 + 0.216i)16-s + (−0.462 + 0.123i)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  :  $\chi_{210} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.932 - 0.360i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.162545 + 0.871698i\)
\(L(\frac12)\)  \(\approx\)  \(0.162545 + 0.871698i\)
\(L(2)\)   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 \( 1 + (-0.626 + 4.96i)T \)
7 \( 1 + (6.84 + 1.45i)T \)
good11 \( 1 + (6.59 - 11.4i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-10.7 + 10.7i)T - 169iT^{2} \)
17 \( 1 + (7.85 - 2.10i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-8.35 + 4.82i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (22.6 + 6.06i)T + (458. + 264.5i)T^{2} \)
29 \( 1 + 41.4iT - 841T^{2} \)
31 \( 1 + (18.0 - 31.3i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-1.25 + 4.68i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 7.31T + 1.68e3T^{2} \)
43 \( 1 + (-48.7 + 48.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (-13.9 + 51.9i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (12.1 + 45.4i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-1.15 - 0.665i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (29.7 + 51.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-117. + 31.5i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 82.6T + 5.04e3T^{2} \)
73 \( 1 + (13.2 + 49.5i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-26.3 + 15.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (32.2 - 32.2i)T - 6.88e3iT^{2} \)
89 \( 1 + (-19.1 + 11.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-128. - 128. i)T + 9.40e3iT^{2} \)
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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

−11.91766956772041476827249986038, −10.58077862882316735057075229391, −9.784649741640390852931532578690, −8.733747758654906069467434514398, −7.62933096284670867178888185019, −6.21885805597717125468444744113, −5.15783463917767733503600273446, −3.80116208915639929157292759504, −2.15437769113024771408171408532, −0.46246401457790067305604490177, 2.97593857603528752309952671321, 3.98643713480416751202912696490, 5.75508066609392566749657737444, 6.28489911469746614010833395620, 7.47357881626348010361696358111, 8.798362660620232375852819228286, 9.670964966604250920961095381882, 10.73745669675572710687071113069, 11.53535474563643032327166853005, 12.91985525653543421861543293042

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