Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.101 + 0.994i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.31 − 1.33i)2-s + (0.256 − 2.98i)3-s + (1.58 − 2.74i)4-s + (1.93 − 1.11i)5-s + (−3.40 − 7.27i)6-s + (4.89 + 5.00i)7-s + 2.23i·8-s + (−8.86 − 1.53i)9-s + (2.99 − 5.18i)10-s + (−15.6 − 9.04i)11-s + (−7.78 − 5.43i)12-s + 2.70·13-s + (18.0 + 5.05i)14-s + (−2.84 − 6.07i)15-s + (9.32 + 16.1i)16-s + (3.39 + 1.96i)17-s + ⋯
L(s)  = 1  + (1.15 − 0.669i)2-s + (0.0855 − 0.996i)3-s + (0.395 − 0.685i)4-s + (0.387 − 0.223i)5-s + (−0.567 − 1.21i)6-s + (0.699 + 0.714i)7-s + 0.279i·8-s + (−0.985 − 0.170i)9-s + (0.299 − 0.518i)10-s + (−1.42 − 0.822i)11-s + (−0.649 − 0.452i)12-s + 0.208·13-s + (1.28 + 0.360i)14-s + (−0.189 − 0.405i)15-s + (0.582 + 1.00i)16-s + (0.199 + 0.115i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.101 + 0.994i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.101 + 0.994i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.81533 - 1.63967i\)
\(L(\frac12)\)  \(\approx\)  \(1.81533 - 1.63967i\)
\(L(2)\)   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 \( 1 + (-0.256 + 2.98i)T \)
5 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 + (-4.89 - 5.00i)T \)
good2 \( 1 + (-2.31 + 1.33i)T + (2 - 3.46i)T^{2} \)
11 \( 1 + (15.6 + 9.04i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 2.70T + 169T^{2} \)
17 \( 1 + (-3.39 - 1.96i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-14.7 - 25.6i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-5.66 + 3.27i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 18.8iT - 841T^{2} \)
31 \( 1 + (-12.6 + 21.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (33.5 + 58.0i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 38.7iT - 1.68e3T^{2} \)
43 \( 1 + 63.9T + 1.84e3T^{2} \)
47 \( 1 + (33.6 - 19.4i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-0.787 - 0.454i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-20.6 - 11.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (25.3 + 43.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (34.7 - 60.1i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 55.2iT - 5.04e3T^{2} \)
73 \( 1 + (-14.8 + 25.7i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (14.9 + 25.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 78.4iT - 6.88e3T^{2} \)
89 \( 1 + (-133. + 77.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 32.8T + 9.40e3T^{2} \)
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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

−13.19011309232447354705869328120, −12.41711934084078982644329281453, −11.60332176830322428695086330116, −10.58541764304722101741610808062, −8.646010538565394179333836814151, −7.82812320883719566351297858597, −5.82981153681469095155957394792, −5.27141233689535077335931880244, −3.17836616448846624532057692152, −1.89728821755233347188819368403, 3.12004363521295084320588607684, 4.75304165787844212989924391798, 5.15879738514034848657028930830, 6.80072591822701033139832099491, 8.021668195219003689274105853926, 9.737577345194951257664924690611, 10.50434721560943399846708650294, 11.73064608401059936186305554545, 13.36071458801750244926523561247, 13.73934195476455350044808521147

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