Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.397 - 0.917i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 − 1.36i)2-s + (0.448 + 1.67i)3-s + (−1.73 − i)4-s + (−4.09 − 2.86i)5-s + 2.44·6-s + (−3.46 + 6.08i)7-s + (−2 + 1.99i)8-s + (−2.59 + 1.50i)9-s + (−5.41 + 4.54i)10-s + (−5.62 + 9.74i)11-s + (0.896 − 3.34i)12-s + (−5.84 + 5.84i)13-s + (7.04 + 6.95i)14-s + (2.96 − 8.13i)15-s + (1.99 + 3.46i)16-s + (24.8 − 6.65i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.149 + 0.557i)3-s + (−0.433 − 0.250i)4-s + (−0.818 − 0.573i)5-s + 0.408·6-s + (−0.494 + 0.869i)7-s + (−0.250 + 0.249i)8-s + (−0.288 + 0.166i)9-s + (−0.541 + 0.454i)10-s + (−0.511 + 0.885i)11-s + (0.0747 − 0.278i)12-s + (−0.449 + 0.449i)13-s + (0.503 + 0.496i)14-s + (0.197 − 0.542i)15-s + (0.124 + 0.216i)16-s + (1.46 − 0.391i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.397 - 0.917i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.397 - 0.917i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.291091 + 0.443372i\)
\(L(\frac12)\)  \(\approx\)  \(0.291091 + 0.443372i\)
\(L(2)\)   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 \( 1 + (-0.366 + 1.36i)T \)
3 \( 1 + (-0.448 - 1.67i)T \)
5 \( 1 + (4.09 + 2.86i)T \)
7 \( 1 + (3.46 - 6.08i)T \)
good11 \( 1 + (5.62 - 9.74i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (5.84 - 5.84i)T - 169iT^{2} \)
17 \( 1 + (-24.8 + 6.65i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (27.5 - 15.8i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (39.8 + 10.6i)T + (458. + 264.5i)T^{2} \)
29 \( 1 - 8.29iT - 841T^{2} \)
31 \( 1 + (-6.00 + 10.3i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-6.32 + 23.6i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 42.9T + 1.68e3T^{2} \)
43 \( 1 + (37.4 - 37.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (-9.60 + 35.8i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (2.96 + 11.0i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-52.5 - 30.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-18.8 - 32.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (78.8 - 21.1i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 1.19T + 5.04e3T^{2} \)
73 \( 1 + (18.5 + 69.0i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (71.0 - 41.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-103. + 103. i)T - 6.88e3iT^{2} \)
89 \( 1 + (12.6 - 7.30i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-24.6 - 24.6i)T + 9.40e3iT^{2} \)
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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

−12.24411568478353845687558511062, −11.81221578546421606129995908806, −10.34824408194547853996553278202, −9.706320686211004810200035971325, −8.649627251355863558242500758653, −7.71237436878018127318452154684, −5.90198840647969335455222493521, −4.73241606801513804695102726015, −3.75803844861075953954089829823, −2.27697072354204286611592098626, 0.26357765092275244725524742312, 3.03868798186374971567182046948, 4.12204864405124740807761048578, 5.82632830367674243460302303593, 6.79619304000068458988590453506, 7.78396363817158108461564054233, 8.284119182368596876370589266542, 9.954129571240342776146489596477, 10.83376860419125288049694057617, 12.05615840174873935913419068058

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