Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.127 + 0.991i$
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 + (4.98 − 0.365i)5-s + 2.44·6-s + (−3.26 − 6.19i)7-s + (−1.99 + 2i)8-s + (2.59 + 1.50i)9-s + (−6.67 + 2.32i)10-s + (−0.187 − 0.325i)11-s + (−3.34 + 0.896i)12-s + (−9.32 + 9.32i)13-s + (6.72 + 7.26i)14-s + (−8.50 − 1.62i)15-s + (1.99 − 3.46i)16-s + (7.97 − 29.7i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.557 − 0.149i)3-s + (0.433 − 0.250i)4-s + (0.997 − 0.0730i)5-s + 0.408·6-s + (−0.466 − 0.884i)7-s + (−0.249 + 0.250i)8-s + (0.288 + 0.166i)9-s + (−0.667 + 0.232i)10-s + (−0.0170 − 0.0295i)11-s + (−0.278 + 0.0747i)12-s + (−0.717 + 0.717i)13-s + (0.480 + 0.518i)14-s + (−0.567 − 0.108i)15-s + (0.124 − 0.216i)16-s + (0.468 − 1.75i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.127 + 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.127 + 0.991i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.127 + 0.991i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.647808 - 0.569796i\)
\(L(\frac12)\)  \(\approx\)  \(0.647808 - 0.569796i\)
\(L(2)\)   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 \( 1 + (-4.98 + 0.365i)T \)
7 \( 1 + (3.26 + 6.19i)T \)
good11 \( 1 + (0.187 + 0.325i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (9.32 - 9.32i)T - 169iT^{2} \)
17 \( 1 + (-7.97 + 29.7i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (-6.46 - 3.73i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (10.3 + 38.4i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + 22.0iT - 841T^{2} \)
31 \( 1 + (23.7 + 41.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (5.90 - 1.58i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 78.1T + 1.68e3T^{2} \)
43 \( 1 + (12.2 - 12.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (11.4 - 3.06i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-3.93 - 1.05i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-2.87 + 1.65i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (44.2 - 76.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-23.1 + 86.4i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 64.3T + 5.04e3T^{2} \)
73 \( 1 + (11.6 + 3.13i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (31.3 + 18.1i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (94.4 - 94.4i)T - 6.88e3iT^{2} \)
89 \( 1 + (50.7 + 29.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-56.0 - 56.0i)T + 9.40e3iT^{2} \)
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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

−11.79043725026088663314830503556, −10.74511960673949381101050313291, −9.800121885767612897630430495621, −9.344753276580881167205412004230, −7.67491449594856110523633659951, −6.83798571893089034085849275641, −5.89862292985143958866190309857, −4.58805965279072843662283891754, −2.44301517031618174998397775820, −0.64762672785903059917842802202, 1.70345254461784551357601521262, 3.24625426562946366278638091943, 5.38600724630749473032003940849, 6.01922500554528884569254502497, 7.29347467081569180977470453008, 8.642255932407852848574107510484, 9.624629423051918819454271711156, 10.23291605619864813790520728533, 11.19777130555953706139102725875, 12.49306237545770261461768624126

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