Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.406 + 0.913i$
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 + (1.86 − 2.35i)3-s + 2i·4-s + (−1.91 − 4.61i)5-s + (4.21 − 0.488i)6-s + (−3.12 − 6.26i)7-s + (−2 + 2i)8-s + (−2.05 − 8.76i)9-s + (2.70 − 6.53i)10-s + 0.117i·11-s + (4.70 + 3.72i)12-s + (9.72 − 9.72i)13-s + (3.13 − 9.39i)14-s + (−14.4 − 4.09i)15-s − 4·16-s + (−13.8 + 13.8i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.620 − 0.783i)3-s + 0.5i·4-s + (−0.383 − 0.923i)5-s + (0.702 − 0.0814i)6-s + (−0.446 − 0.894i)7-s + (−0.250 + 0.250i)8-s + (−0.228 − 0.973i)9-s + (0.270 − 0.653i)10-s + 0.0106i·11-s + (0.391 + 0.310i)12-s + (0.747 − 0.747i)13-s + (0.223 − 0.670i)14-s + (−0.961 − 0.273i)15-s − 0.250·16-s + (−0.815 + 0.815i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.406 + 0.913i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.406 + 0.913i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.74483 - 1.13377i\)
\(L(\frac12)\)  \(\approx\)  \(1.74483 - 1.13377i\)
\(L(2)\)   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 \( 1 + (-1.86 + 2.35i)T \)
5 \( 1 + (1.91 + 4.61i)T \)
7 \( 1 + (3.12 + 6.26i)T \)
good11 \( 1 - 0.117iT - 121T^{2} \)
13 \( 1 + (-9.72 + 9.72i)T - 169iT^{2} \)
17 \( 1 + (13.8 - 13.8i)T - 289iT^{2} \)
19 \( 1 - 29.7T + 361T^{2} \)
23 \( 1 + (2.25 - 2.25i)T - 529iT^{2} \)
29 \( 1 - 46.0T + 841T^{2} \)
31 \( 1 - 1.50iT - 961T^{2} \)
37 \( 1 + (-5.32 + 5.32i)T - 1.36e3iT^{2} \)
41 \( 1 - 13.4T + 1.68e3T^{2} \)
43 \( 1 + (-36.8 - 36.8i)T + 1.84e3iT^{2} \)
47 \( 1 + (29.7 - 29.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (-59.8 + 59.8i)T - 2.80e3iT^{2} \)
59 \( 1 + 84.9iT - 3.48e3T^{2} \)
61 \( 1 - 34.8iT - 3.72e3T^{2} \)
67 \( 1 + (34.4 - 34.4i)T - 4.48e3iT^{2} \)
71 \( 1 - 77.6iT - 5.04e3T^{2} \)
73 \( 1 + (41.3 - 41.3i)T - 5.32e3iT^{2} \)
79 \( 1 - 0.865iT - 6.24e3T^{2} \)
83 \( 1 + (99.0 + 99.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 129. iT - 7.92e3T^{2} \)
97 \( 1 + (-15.7 - 15.7i)T + 9.40e3iT^{2} \)
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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

−12.38366381407589812749604467475, −11.29095916298772717800267672541, −9.780064947840205548614270165835, −8.588605605076954408766283971206, −7.906901614981285128967997545757, −6.93223203884076269500488810063, −5.81503752910077923765507572468, −4.28997929506739774441827161225, −3.21683008583288331384873514386, −1.00475337420811099762516807647, 2.49858969174948579195954744641, 3.34460579354976568873641407872, 4.55874921491193358232792699218, 5.92108344007806780599128769978, 7.17244837669615848622650174125, 8.655189040995032693132984119310, 9.484095726080879454567706142040, 10.43498251791662452635568689569, 11.41260124625142090295430114133, 12.04546637083288962718170391367

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