Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.577 - 0.816i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.50 − 0.869i)2-s + (−1.33 + 2.68i)3-s + (−0.489 + 0.847i)4-s + (1.93 − 1.11i)5-s + (0.325 + 5.20i)6-s + (2.93 + 6.35i)7-s + 8.65i·8-s + (−5.43 − 7.17i)9-s + (1.94 − 3.36i)10-s + (11.0 + 6.38i)11-s + (−1.62 − 2.44i)12-s + 0.690·13-s + (9.93 + 7.01i)14-s + (0.418 + 6.69i)15-s + (5.56 + 9.63i)16-s + (−12.5 − 7.22i)17-s + ⋯
L(s)  = 1  + (0.752 − 0.434i)2-s + (−0.444 + 0.895i)3-s + (−0.122 + 0.211i)4-s + (0.387 − 0.223i)5-s + (0.0542 + 0.867i)6-s + (0.419 + 0.907i)7-s + 1.08i·8-s + (−0.604 − 0.796i)9-s + (0.194 − 0.336i)10-s + (1.00 + 0.580i)11-s + (−0.135 − 0.203i)12-s + 0.0530·13-s + (0.709 + 0.501i)14-s + (0.0279 + 0.446i)15-s + (0.347 + 0.602i)16-s + (−0.735 − 0.424i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.577 - 0.816i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.577 - 0.816i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.52735 + 0.790502i\)
\(L(\frac12)\)  \(\approx\)  \(1.52735 + 0.790502i\)
\(L(2)\)   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 \( 1 + (1.33 - 2.68i)T \)
5 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 + (-2.93 - 6.35i)T \)
good2 \( 1 + (-1.50 + 0.869i)T + (2 - 3.46i)T^{2} \)
11 \( 1 + (-11.0 - 6.38i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 0.690T + 169T^{2} \)
17 \( 1 + (12.5 + 7.22i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (13.8 + 23.9i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-37.3 + 21.5i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 26.6iT - 841T^{2} \)
31 \( 1 + (8.94 - 15.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-17.5 - 30.4i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 15.0iT - 1.68e3T^{2} \)
43 \( 1 + 23.1T + 1.84e3T^{2} \)
47 \( 1 + (-38.1 + 22.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-78.5 - 45.3i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (21.3 + 12.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (33.0 + 57.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (13.6 - 23.5i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 76.5iT - 5.04e3T^{2} \)
73 \( 1 + (-24.7 + 42.8i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-18.5 - 32.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 20.0iT - 6.88e3T^{2} \)
89 \( 1 + (62.7 - 36.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 23.2T + 9.40e3T^{2} \)
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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

−13.57876339886406230245258058354, −12.49031410888949100387858595624, −11.66637434328025936380892720543, −10.87547260939357667462147195951, −9.236755560013827583486615877631, −8.720798734918101016668960678505, −6.51435936032943163678701599324, −5.06291577755742684993675057859, −4.39765375339165535165850316832, −2.64306615349091608482343541155, 1.29963074633768867289222787532, 3.93829752235848479698122079859, 5.45174740620404725243199332595, 6.44846412521882759303436777253, 7.29920463470832880663546914105, 8.886031487177644112716856014277, 10.45640271089196112847887753195, 11.32273367808857398680598906751, 12.72452884369858109738864202607, 13.46721378199116740649330076060

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