Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.953 + 0.302i$
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 − 0.00829i)3-s + 2i·4-s + (−3.67 − 3.39i)5-s + (−2.99 − 3.00i)6-s + (6.84 − 1.45i)7-s + (−2 + 2i)8-s + (8.99 + 0.0497i)9-s + (−0.274 − 7.06i)10-s − 6.08i·11-s + (0.0165 − 5.99i)12-s + (4.00 − 4.00i)13-s + (8.30 + 5.38i)14-s + (10.9 + 10.2i)15-s − 4·16-s + (14.8 − 14.8i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−0.999 − 0.00276i)3-s + 0.5i·4-s + (−0.734 − 0.679i)5-s + (−0.498 − 0.501i)6-s + (0.978 − 0.208i)7-s + (−0.250 + 0.250i)8-s + (0.999 + 0.00553i)9-s + (−0.0274 − 0.706i)10-s − 0.553i·11-s + (0.00138 − 0.499i)12-s + (0.307 − 0.307i)13-s + (0.593 + 0.384i)14-s + (0.732 + 0.681i)15-s − 0.250·16-s + (0.871 − 0.871i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.953 + 0.302i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.953 + 0.302i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.38601 - 0.214887i\)
\(L(\frac12)\)  \(\approx\)  \(1.38601 - 0.214887i\)
\(L(2)\)   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 \( 1 + (2.99 + 0.00829i)T \)
5 \( 1 + (3.67 + 3.39i)T \)
7 \( 1 + (-6.84 + 1.45i)T \)
good11 \( 1 + 6.08iT - 121T^{2} \)
13 \( 1 + (-4.00 + 4.00i)T - 169iT^{2} \)
17 \( 1 + (-14.8 + 14.8i)T - 289iT^{2} \)
19 \( 1 - 20.4T + 361T^{2} \)
23 \( 1 + (-20.6 + 20.6i)T - 529iT^{2} \)
29 \( 1 + 19.5T + 841T^{2} \)
31 \( 1 - 4.36iT - 961T^{2} \)
37 \( 1 + (1.64 - 1.64i)T - 1.36e3iT^{2} \)
41 \( 1 + 42.2T + 1.68e3T^{2} \)
43 \( 1 + (45.0 + 45.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (-36.6 + 36.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (0.652 - 0.652i)T - 2.80e3iT^{2} \)
59 \( 1 - 4.02iT - 3.48e3T^{2} \)
61 \( 1 - 65.2iT - 3.72e3T^{2} \)
67 \( 1 + (-59.7 + 59.7i)T - 4.48e3iT^{2} \)
71 \( 1 - 122. iT - 5.04e3T^{2} \)
73 \( 1 + (13.1 - 13.1i)T - 5.32e3iT^{2} \)
79 \( 1 + 126. iT - 6.24e3T^{2} \)
83 \( 1 + (-12.2 - 12.2i)T + 6.88e3iT^{2} \)
89 \( 1 - 97.2iT - 7.92e3T^{2} \)
97 \( 1 + (-60.6 - 60.6i)T + 9.40e3iT^{2} \)
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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

−11.88630638663599163709230812309, −11.55441225501123416077241460454, −10.42746971303979686006408754229, −8.882802857646069306516035702158, −7.81828585693964891553532918828, −7.01363282554243082650001399344, −5.43154822599783064758130680597, −4.97797681870531575958459534900, −3.66395242453244043157448588819, −0.895985910911315652361091962011, 1.49631843993225844593720479698, 3.51569551699569289954410620008, 4.71097928580318713952182152985, 5.69841504402877993673599993448, 6.99375059400661976580226614407, 7.928395434260459186124614251008, 9.625042964506051451425415367174, 10.66316223664951463651894659011, 11.40288146822998603735680585160, 11.89473409955878758447462492730

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