Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.998 + 0.0515i$
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 + (4.87 − 1.12i)5-s − 2.44·6-s + (6.92 − 1.05i)7-s + (−1.99 + 2i)8-s + (2.59 + 1.50i)9-s + (−6.24 + 3.32i)10-s + (−6.50 − 11.2i)11-s + (3.34 − 0.896i)12-s + (−0.863 + 0.863i)13-s + (−9.06 + 3.96i)14-s + (8.65 + 0.293i)15-s + (1.99 − 3.46i)16-s + (0.241 − 0.902i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.557 + 0.149i)3-s + (0.433 − 0.250i)4-s + (0.974 − 0.225i)5-s − 0.408·6-s + (0.988 − 0.150i)7-s + (−0.249 + 0.250i)8-s + (0.288 + 0.166i)9-s + (−0.624 + 0.332i)10-s + (−0.591 − 1.02i)11-s + (0.278 − 0.0747i)12-s + (−0.0664 + 0.0664i)13-s + (−0.647 + 0.283i)14-s + (0.577 + 0.0195i)15-s + (0.124 − 0.216i)16-s + (0.0142 − 0.0531i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0515i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.998 + 0.0515i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.998 + 0.0515i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.69292 - 0.0436599i\)
\(L(\frac12)\)  \(\approx\)  \(1.69292 - 0.0436599i\)
\(L(2)\)   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 \( 1 + (-4.87 + 1.12i)T \)
7 \( 1 + (-6.92 + 1.05i)T \)
good11 \( 1 + (6.50 + 11.2i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (0.863 - 0.863i)T - 169iT^{2} \)
17 \( 1 + (-0.241 + 0.902i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (-5.84 - 3.37i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (3.90 + 14.5i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 - 31.3iT - 841T^{2} \)
31 \( 1 + (-23.4 - 40.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-36.5 + 9.80i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 64.3T + 1.68e3T^{2} \)
43 \( 1 + (-17.7 + 17.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (-48.1 + 12.9i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (72.2 + 19.3i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (31.0 - 17.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (54.5 - 94.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (16.2 - 60.4i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 74.7T + 5.04e3T^{2} \)
73 \( 1 + (80.2 + 21.5i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (83.9 + 48.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-31.9 + 31.9i)T - 6.88e3iT^{2} \)
89 \( 1 + (-44.1 - 25.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (83.5 + 83.5i)T + 9.40e3iT^{2} \)
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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

−12.02714516280938735506066310781, −10.74691818106018435404699839723, −10.22358854457921718335053486195, −8.913866900179727458486885495582, −8.434443889677886966097596412068, −7.28449471656844975193185608201, −5.91113718416832451320942413032, −4.83184311793245522737974874573, −2.84576115673150218306704811908, −1.39077222409628447125624509915, 1.66991945306836982563838329990, 2.66561292417913969455265770596, 4.65216009923947496836723361646, 6.05111946736187799903348005130, 7.42633350950142905998889961399, 8.110094035677589953209667841901, 9.384071647320020367958213507269, 9.950318951704276416986733075846, 11.01412870177159396372212308721, 12.05470566533150900070266228772

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