Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.578 + 0.815i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.897 − 1.55i)2-s + (2.91 + 0.725i)3-s + (0.387 − 0.671i)4-s + (4.89 − 0.996i)5-s + (−1.48 − 5.17i)6-s + (1.39 + 6.85i)7-s − 8.57·8-s + (7.94 + 4.22i)9-s + (−5.94 − 6.72i)10-s + (−1.00 − 0.581i)11-s + (1.61 − 1.67i)12-s − 12.4i·13-s + (9.41 − 8.32i)14-s + (14.9 + 0.652i)15-s + (6.14 + 10.6i)16-s + (−9.31 + 16.1i)17-s + ⋯
L(s)  = 1  + (−0.448 − 0.777i)2-s + (0.970 + 0.241i)3-s + (0.0969 − 0.167i)4-s + (0.979 − 0.199i)5-s + (−0.247 − 0.862i)6-s + (0.199 + 0.979i)7-s − 1.07·8-s + (0.883 + 0.469i)9-s + (−0.594 − 0.672i)10-s + (−0.0916 − 0.0528i)11-s + (0.134 − 0.139i)12-s − 0.959i·13-s + (0.672 − 0.594i)14-s + (0.999 + 0.0435i)15-s + (0.384 + 0.665i)16-s + (−0.547 + 0.949i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.578 + 0.815i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (74, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.578 + 0.815i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.48281 - 0.765735i\)
\(L(\frac12)\)  \(\approx\)  \(1.48281 - 0.765735i\)
\(L(2)\)   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 \( 1 + (-2.91 - 0.725i)T \)
5 \( 1 + (-4.89 + 0.996i)T \)
7 \( 1 + (-1.39 - 6.85i)T \)
good2 \( 1 + (0.897 + 1.55i)T + (-2 + 3.46i)T^{2} \)
11 \( 1 + (1.00 + 0.581i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 12.4iT - 169T^{2} \)
17 \( 1 + (9.31 - 16.1i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (15.2 + 26.3i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (13.7 + 23.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 52.6iT - 841T^{2} \)
31 \( 1 + (17.2 - 29.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (0.357 - 0.206i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 17.2iT - 1.68e3T^{2} \)
43 \( 1 - 7.86iT - 1.84e3T^{2} \)
47 \( 1 + (17.4 + 30.1i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-17.8 + 30.9i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-32.3 - 18.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-25.4 - 44.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (24.9 + 14.4i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 66.8iT - 5.04e3T^{2} \)
73 \( 1 + (46.7 + 27.0i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (16.6 + 28.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 72.0T + 6.88e3T^{2} \)
89 \( 1 + (-41.4 + 23.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 66.7iT - 9.40e3T^{2} \)
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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

−13.13338885319902572061015171515, −12.48954262414029083730449219198, −10.81573673118314074032984975167, −10.22008698220102635084540733267, −8.933387788513255284177228482536, −8.628231017270204095588760485964, −6.49671571604867399737231144354, −5.11863755903280364931864308736, −2.88921176636023085060314910451, −1.88334839087800448039305854121, 2.15025984283796815906756053979, 3.95447309464479895842636593722, 6.16292354217262581689245640396, 7.16701312046865336427470534161, 8.018321598115251678998057835364, 9.253132377054877141838246187502, 9.996233884482719489531707317720, 11.60652468813583421639813335392, 13.03579463933496061497625896833, 13.89038191974460571917753523395

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