Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.983 - 0.178i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 − 0.366i)2-s + (1.67 + 0.448i)3-s + (1.73 − i)4-s + (4.73 + 1.61i)5-s + 2.44·6-s + (−4.13 + 5.65i)7-s + (1.99 − 2i)8-s + (2.59 + 1.50i)9-s + (7.05 + 0.479i)10-s + (2.47 + 4.28i)11-s + (3.34 − 0.896i)12-s + (7.82 − 7.82i)13-s + (−3.57 + 9.23i)14-s + (7.18 + 4.82i)15-s + (1.99 − 3.46i)16-s + (−0.914 + 3.41i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.557 + 0.149i)3-s + (0.433 − 0.250i)4-s + (0.946 + 0.323i)5-s + 0.408·6-s + (−0.590 + 0.807i)7-s + (0.249 − 0.250i)8-s + (0.288 + 0.166i)9-s + (0.705 + 0.0479i)10-s + (0.224 + 0.389i)11-s + (0.278 − 0.0747i)12-s + (0.602 − 0.602i)13-s + (−0.255 + 0.659i)14-s + (0.479 + 0.321i)15-s + (0.124 − 0.216i)16-s + (−0.0537 + 0.200i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.983 - 0.178i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.983 - 0.178i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.92638 + 0.263194i\)
\(L(\frac12)\)  \(\approx\)  \(2.92638 + 0.263194i\)
\(L(2)\)   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 \( 1 + (-1.36 + 0.366i)T \)
3 \( 1 + (-1.67 - 0.448i)T \)
5 \( 1 + (-4.73 - 1.61i)T \)
7 \( 1 + (4.13 - 5.65i)T \)
good11 \( 1 + (-2.47 - 4.28i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-7.82 + 7.82i)T - 169iT^{2} \)
17 \( 1 + (0.914 - 3.41i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (26.8 + 15.5i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (4.61 + 17.2i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + 24.0iT - 841T^{2} \)
31 \( 1 + (-7.79 - 13.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-33.2 + 8.92i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 19.3T + 1.68e3T^{2} \)
43 \( 1 + (11.5 - 11.5i)T - 1.84e3iT^{2} \)
47 \( 1 + (30.3 - 8.13i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (100. + 26.9i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (13.5 - 7.81i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (44.1 - 76.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-20.2 + 75.6i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 59.7T + 5.04e3T^{2} \)
73 \( 1 + (-102. - 27.5i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-23.6 - 13.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (48.9 - 48.9i)T - 6.88e3iT^{2} \)
89 \( 1 + (98.6 + 56.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-99.3 - 99.3i)T + 9.40e3iT^{2} \)
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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

−12.57861620894492291320747641781, −11.15511225380663366052104482125, −10.21941406439470013913236654890, −9.349382410560223943929219362317, −8.326987837966818178888123618146, −6.64915799778922019379092980937, −5.99979104595018757501272234498, −4.61445706687072356274066045695, −3.09801183989831215736353045520, −2.10792401412545052848177456450, 1.69037018586905380950111495725, 3.33653248451353459104716580684, 4.47566192712798484716081722508, 6.04268086191504000086113296576, 6.70229732092095069209750013129, 8.065202232910514077079882419401, 9.155365528918896592882579486152, 10.11927355774276340409992836098, 11.16952822997445858784598000969, 12.55700767296915324598435820661

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