Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.582 + 0.813i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.336 − 0.583i)2-s + (−1.5 − 0.866i)3-s + (1.77 − 3.07i)4-s + (1.93 − 1.11i)5-s + 1.16i·6-s + (−6.82 − 1.55i)7-s − 5.08·8-s + (1.5 + 2.59i)9-s + (−1.30 − 0.752i)10-s + (0.0223 − 0.0387i)11-s + (−5.31 + 3.07i)12-s − 23.0i·13-s + (1.38 + 4.50i)14-s − 3.87·15-s + (−5.38 − 9.32i)16-s + (−8.16 − 4.71i)17-s + ⋯
L(s)  = 1  + (−0.168 − 0.291i)2-s + (−0.5 − 0.288i)3-s + (0.443 − 0.767i)4-s + (0.387 − 0.223i)5-s + 0.194i·6-s + (−0.974 − 0.222i)7-s − 0.635·8-s + (0.166 + 0.288i)9-s + (−0.130 − 0.0752i)10-s + (0.00203 − 0.00352i)11-s + (−0.443 + 0.255i)12-s − 1.76i·13-s + (0.0992 + 0.321i)14-s − 0.258·15-s + (−0.336 − 0.582i)16-s + (−0.480 − 0.277i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 + 0.813i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.582 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.582 + 0.813i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.582 + 0.813i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.452897 - 0.881111i\)
\(L(\frac12)\)  \(\approx\)  \(0.452897 - 0.881111i\)
\(L(2)\)   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 \( 1 + (6.82 + 1.55i)T \)
good2 \( 1 + (0.336 + 0.583i)T + (-2 + 3.46i)T^{2} \)
11 \( 1 + (-0.0223 + 0.0387i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 23.0iT - 169T^{2} \)
17 \( 1 + (8.16 + 4.71i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-0.991 + 0.572i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-22.1 - 38.3i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 53.0T + 841T^{2} \)
31 \( 1 + (-19.5 - 11.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (21.1 + 36.6i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 38.2iT - 1.68e3T^{2} \)
43 \( 1 - 76.5T + 1.84e3T^{2} \)
47 \( 1 + (23.5 - 13.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (9.49 - 16.4i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (4.21 + 2.43i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (33.6 - 19.4i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-3.50 + 6.06i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 46.8T + 5.04e3T^{2} \)
73 \( 1 + (-72.3 - 41.7i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (10.2 + 17.7i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 125. iT - 6.88e3T^{2} \)
89 \( 1 + (-40.4 + 23.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 3.11iT - 9.40e3T^{2} \)
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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

−13.02618650316160683018610898674, −12.14094765231524895269043659498, −10.82522314945814281282932898833, −10.17988342292801541553202273630, −9.115463375122662790652720564908, −7.34253148671810718466428833513, −6.18021160223615948083412274311, −5.27292227560566692531178341960, −2.89933062494391018806016300811, −0.833142826775340788140930803211, 2.72573715326276973194961801925, 4.40275203480249749543121459282, 6.48317050323161396664016920623, 6.67085253903701881664410209718, 8.554385060547114977990262329105, 9.512620473547530742316341924255, 10.76270634359486228078570194919, 11.87294970163043742152201354515, 12.65063709604533991501082533005, 13.84507572225060753713825768832

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