Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.728 - 0.685i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.18 + 0.853i)2-s + (1.67 + 0.448i)3-s + (5.95 − 3.43i)4-s + (−4.91 + 0.932i)5-s − 5.71·6-s + (5.00 + 4.89i)7-s + (−6.69 + 6.69i)8-s + (2.59 + 1.50i)9-s + (14.8 − 7.16i)10-s + (−0.581 − 1.00i)11-s + (11.4 − 3.08i)12-s + (−14.6 + 14.6i)13-s + (−20.1 − 11.3i)14-s + (−8.63 − 0.641i)15-s + (1.87 − 3.23i)16-s + (−6.97 + 26.0i)17-s + ⋯
L(s)  = 1  + (−1.59 + 0.426i)2-s + (0.557 + 0.149i)3-s + (1.48 − 0.859i)4-s + (−0.982 + 0.186i)5-s − 0.951·6-s + (0.714 + 0.699i)7-s + (−0.837 + 0.837i)8-s + (0.288 + 0.166i)9-s + (1.48 − 0.716i)10-s + (−0.0529 − 0.0916i)11-s + (0.958 − 0.256i)12-s + (−1.13 + 1.13i)13-s + (−1.43 − 0.808i)14-s + (−0.575 − 0.0427i)15-s + (0.116 − 0.202i)16-s + (−0.410 + 1.53i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.685i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.728 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.728 - 0.685i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.728 - 0.685i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.181090 + 0.456685i\)
\(L(\frac12)\)  \(\approx\)  \(0.181090 + 0.456685i\)
\(L(2)\)   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 \( 1 + (4.91 - 0.932i)T \)
7 \( 1 + (-5.00 - 4.89i)T \)
good2 \( 1 + (3.18 - 0.853i)T + (3.46 - 2i)T^{2} \)
11 \( 1 + (0.581 + 1.00i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (14.6 - 14.6i)T - 169iT^{2} \)
17 \( 1 + (6.97 - 26.0i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (24.9 + 14.4i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-3.19 - 11.9i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 - 12.5iT - 841T^{2} \)
31 \( 1 + (-8.24 - 14.2i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-13.6 + 3.66i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 48.3T + 1.68e3T^{2} \)
43 \( 1 + (0.357 - 0.357i)T - 1.84e3iT^{2} \)
47 \( 1 + (-27.3 + 7.33i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (45.5 + 12.2i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-49.5 + 28.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-6.30 + 10.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-12.0 + 44.9i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 19.0T + 5.04e3T^{2} \)
73 \( 1 + (118. + 31.7i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-74.1 - 42.8i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-18.0 + 18.0i)T - 6.88e3iT^{2} \)
89 \( 1 + (78.1 + 45.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-84.2 - 84.2i)T + 9.40e3iT^{2} \)
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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

−14.55997595515178845224593522004, −12.64497962970376831895715514634, −11.39247341931301308629060477937, −10.58901370483988473951950000856, −9.196112804191258950433370301143, −8.534877463124727551903381409273, −7.68009224981334320016418402630, −6.60204025667589207101078312792, −4.41515898080799762352034448433, −2.11369563383219280222401989805, 0.55995182012661999440141329018, 2.57976503869034825093836418080, 4.51157702803520032796852931331, 7.23775620943983195762609232372, 7.80583555224977683235116946772, 8.634173536039418275769328739500, 9.866076660656482238447017793479, 10.78969791178770492236678497505, 11.74484966035885002726627138698, 12.80060871473924916444525409623

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