Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.310 + 0.950i$
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 + (1.22 − 1.22i)3-s + 2i·4-s + (4.26 − 2.60i)5-s − 2.44·6-s + (1.87 + 1.87i)7-s + (2 − 2i)8-s − 2.99i·9-s + (−6.87 − 1.66i)10-s + 11.5·11-s + (2.44 + 2.44i)12-s + (−2.01 + 2.01i)13-s − 3.74i·14-s + (2.04 − 8.41i)15-s − 4·16-s + (7.75 + 7.75i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (0.853 − 0.520i)5-s − 0.408·6-s + (0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.687 − 0.166i)10-s + 1.05·11-s + (0.204 + 0.204i)12-s + (−0.155 + 0.155i)13-s − 0.267i·14-s + (0.136 − 0.561i)15-s − 0.250·16-s + (0.456 + 0.456i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.310 + 0.950i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (127, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.310 + 0.950i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.33438 - 0.967871i\)
\(L(\frac12)\)  \(\approx\)  \(1.33438 - 0.967871i\)
\(L(2)\)   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 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 + (-4.26 + 2.60i)T \)
7 \( 1 + (-1.87 - 1.87i)T \)
good11 \( 1 - 11.5T + 121T^{2} \)
13 \( 1 + (2.01 - 2.01i)T - 169iT^{2} \)
17 \( 1 + (-7.75 - 7.75i)T + 289iT^{2} \)
19 \( 1 + 21.8iT - 361T^{2} \)
23 \( 1 + (7.41 - 7.41i)T - 529iT^{2} \)
29 \( 1 + 31.0iT - 841T^{2} \)
31 \( 1 + 11.5T + 961T^{2} \)
37 \( 1 + (32.0 + 32.0i)T + 1.36e3iT^{2} \)
41 \( 1 - 39.6T + 1.68e3T^{2} \)
43 \( 1 + (19.3 - 19.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (-21.8 - 21.8i)T + 2.20e3iT^{2} \)
53 \( 1 + (42.4 - 42.4i)T - 2.80e3iT^{2} \)
59 \( 1 - 89.0iT - 3.48e3T^{2} \)
61 \( 1 - 7.07T + 3.72e3T^{2} \)
67 \( 1 + (-15.6 - 15.6i)T + 4.48e3iT^{2} \)
71 \( 1 - 133.T + 5.04e3T^{2} \)
73 \( 1 + (92.3 - 92.3i)T - 5.32e3iT^{2} \)
79 \( 1 - 126. iT - 6.24e3T^{2} \)
83 \( 1 + (30.0 - 30.0i)T - 6.88e3iT^{2} \)
89 \( 1 - 4.93iT - 7.92e3T^{2} \)
97 \( 1 + (30.2 + 30.2i)T + 9.40e3iT^{2} \)
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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

−12.02467894596991120831636432827, −11.00170578860315062966478164106, −9.682913300740965758027353573373, −9.133910154896714984421447693786, −8.219009342589847556362326158635, −6.96032737694361188736390580007, −5.74598428652444534795028695545, −4.17344231096642841782680404335, −2.47425528455941239649156362507, −1.25736055090861230815544012856, 1.71009064446271767592586966835, 3.49068652235121246833666757195, 5.09761096536803809239693154799, 6.27675981576519263789582057752, 7.27992493614826057610857360367, 8.444361509552160388712121343659, 9.454777448598201781730165178436, 10.12951379945265843272954847706, 11.01822163225181339095987321589, 12.30439108405937700428930487154

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