Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.938 + 0.343i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.836 − 1.44i)2-s + (−1.5 + 0.866i)3-s + (0.599 + 1.03i)4-s + (1.93 + 1.11i)5-s + 2.89i·6-s + (4.76 − 5.13i)7-s + 8.70·8-s + (1.5 − 2.59i)9-s + (3.24 − 1.87i)10-s + (6.91 + 11.9i)11-s + (−1.79 − 1.03i)12-s − 6.12i·13-s + (−3.45 − 11.1i)14-s − 3.87·15-s + (4.88 − 8.45i)16-s + (−2.14 + 1.23i)17-s + ⋯
L(s)  = 1  + (0.418 − 0.724i)2-s + (−0.5 + 0.288i)3-s + (0.149 + 0.259i)4-s + (0.387 + 0.223i)5-s + 0.483i·6-s + (0.680 − 0.733i)7-s + 1.08·8-s + (0.166 − 0.288i)9-s + (0.324 − 0.187i)10-s + (0.628 + 1.08i)11-s + (−0.149 − 0.0865i)12-s − 0.470i·13-s + (−0.246 − 0.799i)14-s − 0.258·15-s + (0.305 − 0.528i)16-s + (−0.126 + 0.0729i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.343i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.938 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.938 + 0.343i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (61, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.938 + 0.343i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.71214 - 0.303708i\)
\(L(\frac12)\)  \(\approx\)  \(1.71214 - 0.303708i\)
\(L(2)\)   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 \( 1 + (1.5 - 0.866i)T \)
5 \( 1 + (-1.93 - 1.11i)T \)
7 \( 1 + (-4.76 + 5.13i)T \)
good2 \( 1 + (-0.836 + 1.44i)T + (-2 - 3.46i)T^{2} \)
11 \( 1 + (-6.91 - 11.9i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 6.12iT - 169T^{2} \)
17 \( 1 + (2.14 - 1.23i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (24.2 + 13.9i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (6.62 - 11.4i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 27.6T + 841T^{2} \)
31 \( 1 + (16.2 - 9.36i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-20.5 + 35.5i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 22.5iT - 1.68e3T^{2} \)
43 \( 1 - 7.60T + 1.84e3T^{2} \)
47 \( 1 + (11.9 + 6.88i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (46.2 + 80.1i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (61.5 - 35.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (100. + 57.8i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-5.70 - 9.87i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 99.4T + 5.04e3T^{2} \)
73 \( 1 + (-90.1 + 52.0i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (64.4 - 111. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 30.3iT - 6.88e3T^{2} \)
89 \( 1 + (-93.9 - 54.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 153. iT - 9.40e3T^{2} \)
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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

−13.18835814098306043709317036267, −12.40977972178170532468662290306, −11.21310423487082880499740846104, −10.70609186134745956158946474236, −9.526420598272847924289957469071, −7.77236658476725703705825999127, −6.68696481755032347903305224215, −4.88615911383379877913401407471, −3.86134600669325916665439884792, −1.90301847333405638741661081241, 1.75432148002059512357107504946, 4.50951009309382228806057501457, 5.81308372160759255613955500692, 6.34942347181675341267424712318, 7.888654117212951118666090471352, 9.073052249086132167982166868391, 10.65194597565689597422400418597, 11.47557539133000075610250134852, 12.64897936477235570182028134126, 13.84311989826912910723472374565

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