Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.999 - 0.0208i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.13 − 0.571i)2-s + (1.67 + 0.448i)3-s + (0.756 − 0.436i)4-s + (2.78 + 4.15i)5-s + 3.82·6-s + (2.73 − 6.44i)7-s + (−4.88 + 4.88i)8-s + (2.59 + 1.50i)9-s + (8.31 + 7.26i)10-s + (−4.69 − 8.12i)11-s + (1.46 − 0.391i)12-s + (−0.405 + 0.405i)13-s + (2.15 − 15.3i)14-s + (2.79 + 8.19i)15-s + (−9.36 + 16.2i)16-s + (2.82 − 10.5i)17-s + ⋯
L(s)  = 1  + (1.06 − 0.285i)2-s + (0.557 + 0.149i)3-s + (0.189 − 0.109i)4-s + (0.557 + 0.830i)5-s + 0.637·6-s + (0.390 − 0.920i)7-s + (−0.610 + 0.610i)8-s + (0.288 + 0.166i)9-s + (0.831 + 0.726i)10-s + (−0.426 − 0.738i)11-s + (0.121 − 0.0326i)12-s + (−0.0311 + 0.0311i)13-s + (0.153 − 1.09i)14-s + (0.186 + 0.546i)15-s + (−0.585 + 1.01i)16-s + (0.166 − 0.621i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0208i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.999 - 0.0208i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.999 - 0.0208i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.53853 + 0.0264045i\)
\(L(\frac12)\)  \(\approx\)  \(2.53853 + 0.0264045i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (-1.67 - 0.448i)T \)
5 \( 1 + (-2.78 - 4.15i)T \)
7 \( 1 + (-2.73 + 6.44i)T \)
good2 \( 1 + (-2.13 + 0.571i)T + (3.46 - 2i)T^{2} \)
11 \( 1 + (4.69 + 8.12i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (0.405 - 0.405i)T - 169iT^{2} \)
17 \( 1 + (-2.82 + 10.5i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (23.8 + 13.7i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-1.02 - 3.81i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 - 34.6iT - 841T^{2} \)
31 \( 1 + (11.8 + 20.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-25.6 + 6.86i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 54.6T + 1.68e3T^{2} \)
43 \( 1 + (47.3 - 47.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (16.9 - 4.55i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-14.7 - 3.95i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-90.3 + 52.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (12.5 - 21.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-0.364 + 1.36i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 86.5T + 5.04e3T^{2} \)
73 \( 1 + (84.9 + 22.7i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-128. - 74.1i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-35.5 + 35.5i)T - 6.88e3iT^{2} \)
89 \( 1 + (-130. - 75.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-74.8 - 74.8i)T + 9.40e3iT^{2} \)
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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

−13.50246991452089202055315867169, −13.01805072148110587464693684759, −11.32677890425682963801255767493, −10.67149255781752636329695409323, −9.290618626535639528014095854400, −7.947309328770989942758243035942, −6.56129284206556171233112386540, −5.09340424575573959151660763335, −3.76436366736412027314598896122, −2.56529055245410112651385257442, 2.19096069793402111425857903784, 4.17887507140922445692307541654, 5.29505859368181476822462418780, 6.30902853456022444505732977489, 8.100410605313597915033677441780, 9.073977444701743223464930451672, 10.13837553257817061352344557104, 12.08437932718170273558435193368, 12.72584334838124959036019772543, 13.43127739647490963329124145235

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