Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.121 - 0.992i$
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 + (−4.19 + 2.72i)5-s − 2.44·6-s + (−1.84 + 6.75i)7-s + (1.99 − 2i)8-s + (2.59 + 1.50i)9-s + (−4.73 + 5.25i)10-s + (7.12 + 12.3i)11-s + (−3.34 + 0.896i)12-s + (−2.75 + 2.75i)13-s + (−0.0509 + 9.89i)14-s + (8.23 − 2.67i)15-s + (1.99 − 3.46i)16-s + (−6.41 + 23.9i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.557 − 0.149i)3-s + (0.433 − 0.250i)4-s + (−0.839 + 0.544i)5-s − 0.408·6-s + (−0.263 + 0.964i)7-s + (0.249 − 0.250i)8-s + (0.288 + 0.166i)9-s + (−0.473 + 0.525i)10-s + (0.647 + 1.12i)11-s + (−0.278 + 0.0747i)12-s + (−0.212 + 0.212i)13-s + (−0.00363 + 0.707i)14-s + (0.549 − 0.178i)15-s + (0.124 − 0.216i)16-s + (−0.377 + 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.121 - 0.992i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.121 - 0.992i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.00884 + 0.892733i\)
\(L(\frac12)\)  \(\approx\)  \(1.00884 + 0.892733i\)
\(L(2)\)   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 \( 1 + (4.19 - 2.72i)T \)
7 \( 1 + (1.84 - 6.75i)T \)
good11 \( 1 + (-7.12 - 12.3i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (2.75 - 2.75i)T - 169iT^{2} \)
17 \( 1 + (6.41 - 23.9i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (-0.277 - 0.159i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (3.64 + 13.6i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 - 24.2iT - 841T^{2} \)
31 \( 1 + (7.62 + 13.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (2.28 - 0.611i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 29.0T + 1.68e3T^{2} \)
43 \( 1 + (11.7 - 11.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (81.7 - 21.9i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-72.6 - 19.4i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-31.7 + 18.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-54.2 + 93.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-17.6 + 66.0i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 20.3T + 5.04e3T^{2} \)
73 \( 1 + (-56.5 - 15.1i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-44.5 - 25.7i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (82.8 - 82.8i)T - 6.88e3iT^{2} \)
89 \( 1 + (-95.6 - 55.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-68.2 - 68.2i)T + 9.40e3iT^{2} \)
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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

−12.39342598908750884468833852666, −11.60025013888906814081454528729, −10.75237665290453008919787372108, −9.630771262203368845380559911526, −8.234140673657237023003841391943, −6.91885847479919911956332433975, −6.23651636801542504114227012058, −4.83356360587721801753291458462, −3.73346799250853268303930439201, −2.11127566577221693337121489452, 0.65554813988911445324829791987, 3.40603326239116368241919732792, 4.35087056385769628919362390950, 5.44737822216614751765648576482, 6.74469700321268135039431883510, 7.60287235328899683939233826370, 8.864735972431658006401867893439, 10.15614710733396553190921430746, 11.54429954087610144804607778681, 11.57035132178237894929035346852

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