Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.516 - 0.856i$
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 + (2.99 + 0.199i)3-s − 2i·4-s + (−4.37 + 2.41i)5-s + (−3.19 + 2.79i)6-s + (4.76 − 5.12i)7-s + (2 + 2i)8-s + (8.92 + 1.19i)9-s + (1.95 − 6.79i)10-s + 6.70i·11-s + (0.398 − 5.98i)12-s + (16.0 + 16.0i)13-s + (0.359 + 9.89i)14-s + (−13.5 + 6.36i)15-s − 4·16-s + (7.21 + 7.21i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.997 + 0.0664i)3-s − 0.5i·4-s + (−0.875 + 0.483i)5-s + (−0.532 + 0.465i)6-s + (0.680 − 0.732i)7-s + (0.250 + 0.250i)8-s + (0.991 + 0.132i)9-s + (0.195 − 0.679i)10-s + 0.609i·11-s + (0.0332 − 0.498i)12-s + (1.23 + 1.23i)13-s + (0.0256 + 0.706i)14-s + (−0.905 + 0.424i)15-s − 0.250·16-s + (0.424 + 0.424i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.516 - 0.856i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (167, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.516 - 0.856i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.40918 + 0.795918i\)
\(L(\frac12)\)  \(\approx\)  \(1.40918 + 0.795918i\)
\(L(2)\)   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 \( 1 + (-2.99 - 0.199i)T \)
5 \( 1 + (4.37 - 2.41i)T \)
7 \( 1 + (-4.76 + 5.12i)T \)
good11 \( 1 - 6.70iT - 121T^{2} \)
13 \( 1 + (-16.0 - 16.0i)T + 169iT^{2} \)
17 \( 1 + (-7.21 - 7.21i)T + 289iT^{2} \)
19 \( 1 - 8.06T + 361T^{2} \)
23 \( 1 + (11.7 + 11.7i)T + 529iT^{2} \)
29 \( 1 + 6.17T + 841T^{2} \)
31 \( 1 - 41.4iT - 961T^{2} \)
37 \( 1 + (37.8 + 37.8i)T + 1.36e3iT^{2} \)
41 \( 1 - 74.2T + 1.68e3T^{2} \)
43 \( 1 + (42.3 - 42.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (39.4 + 39.4i)T + 2.20e3iT^{2} \)
53 \( 1 + (-44.4 - 44.4i)T + 2.80e3iT^{2} \)
59 \( 1 + 51.9iT - 3.48e3T^{2} \)
61 \( 1 + 15.0iT - 3.72e3T^{2} \)
67 \( 1 + (38.7 + 38.7i)T + 4.48e3iT^{2} \)
71 \( 1 + 128. iT - 5.04e3T^{2} \)
73 \( 1 + (54.2 + 54.2i)T + 5.32e3iT^{2} \)
79 \( 1 + 25.7iT - 6.24e3T^{2} \)
83 \( 1 + (27.7 - 27.7i)T - 6.88e3iT^{2} \)
89 \( 1 + 32.5iT - 7.92e3T^{2} \)
97 \( 1 + (56.7 - 56.7i)T - 9.40e3iT^{2} \)
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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

−12.24731089865595413893905916134, −11.05372161238410667688442088144, −10.31075493834208450596859639492, −9.078791205182414086257419092991, −8.209647926271218109982317147874, −7.44256923822897453215396151158, −6.61250685223320912752579637831, −4.55837310241491396947685195697, −3.62839133528922676218756696223, −1.62286525221031833680702596932, 1.17757079519430826686815939141, 2.94030044147326896623160772239, 3.96489444019646370623471628809, 5.55612216975466202840567728687, 7.50613461799461303514360130389, 8.269780001729170358621276473778, 8.720108851055914325189334942020, 9.879553427652693664272891535482, 11.13171861231084682319000743262, 11.86548492991634128535363574537

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