Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.784 - 0.620i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.987 − 0.569i)2-s + (−0.202 + 2.99i)3-s + (−1.35 − 2.33i)4-s + (−1.93 − 1.11i)5-s + (1.90 − 2.83i)6-s + (2.61 + 6.49i)7-s + 7.63i·8-s + (−8.91 − 1.21i)9-s + (1.27 + 2.20i)10-s + (−12.8 + 7.41i)11-s + (7.27 − 3.56i)12-s − 23.9·13-s + (1.12 − 7.89i)14-s + (3.73 − 5.56i)15-s + (−1.04 + 1.81i)16-s + (5.54 − 3.20i)17-s + ⋯
L(s)  = 1  + (−0.493 − 0.284i)2-s + (−0.0674 + 0.997i)3-s + (−0.337 − 0.584i)4-s + (−0.387 − 0.223i)5-s + (0.317 − 0.473i)6-s + (0.372 + 0.927i)7-s + 0.954i·8-s + (−0.990 − 0.134i)9-s + (0.127 + 0.220i)10-s + (−1.16 + 0.673i)11-s + (0.606 − 0.297i)12-s − 1.84·13-s + (0.0803 − 0.564i)14-s + (0.249 − 0.371i)15-s + (−0.0655 + 0.113i)16-s + (0.326 − 0.188i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.784 - 0.620i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (86, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.784 - 0.620i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.125307 + 0.360428i\)
\(L(\frac12)\)  \(\approx\)  \(0.125307 + 0.360428i\)
\(L(2)\)   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 \( 1 + (0.202 - 2.99i)T \)
5 \( 1 + (1.93 + 1.11i)T \)
7 \( 1 + (-2.61 - 6.49i)T \)
good2 \( 1 + (0.987 + 0.569i)T + (2 + 3.46i)T^{2} \)
11 \( 1 + (12.8 - 7.41i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 23.9T + 169T^{2} \)
17 \( 1 + (-5.54 + 3.20i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-3.55 + 6.16i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-28.8 - 16.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 32.6iT - 841T^{2} \)
31 \( 1 + (1.00 + 1.73i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-9.06 + 15.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 40.8iT - 1.68e3T^{2} \)
43 \( 1 + 29.7T + 1.84e3T^{2} \)
47 \( 1 + (-6.22 - 3.59i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (34.1 - 19.7i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (66.6 - 38.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-13.9 + 24.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-50.9 - 88.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 65.1iT - 5.04e3T^{2} \)
73 \( 1 + (-58.9 - 102. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-17.0 + 29.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 34.5iT - 6.88e3T^{2} \)
89 \( 1 + (28.7 + 16.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 5.32T + 9.40e3T^{2} \)
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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

−14.32126278104311709764796487755, −12.64232974518811013877444100815, −11.59035022039400677218191687289, −10.57375785522631412141902862308, −9.633239696165077932202387464353, −8.916766028425300903731014892732, −7.61415663575386098893808598711, −5.20370793727779653310592325599, −4.99515486640588147194497981606, −2.59615529657076163736618621786, 0.32336140297977028702953172164, 2.97082726165237751777158084461, 4.87517349067993007650083601252, 6.79521830875615764359924061358, 7.73230839241965739739210105873, 8.156248502557994654567458139352, 9.814310006311564268789414936398, 11.05538416024671611043542587178, 12.24959337923948627531222422357, 13.06010906963502419685878063417

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