Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.963 - 0.267i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.675 − 0.675i)2-s + (1.22 + 1.22i)3-s + 3.08i·4-s + (3.39 − 3.67i)5-s + 1.65·6-s + (−1.87 + 1.87i)7-s + (4.78 + 4.78i)8-s + 2.99i·9-s + (−0.186 − 4.77i)10-s + 7.59·11-s + (−3.78 + 3.78i)12-s + (1.12 + 1.12i)13-s + 2.52i·14-s + (8.65 − 0.337i)15-s − 5.88·16-s + (−3.43 + 3.43i)17-s + ⋯
L(s)  = 1  + (0.337 − 0.337i)2-s + (0.408 + 0.408i)3-s + 0.771i·4-s + (0.678 − 0.734i)5-s + 0.275·6-s + (−0.267 + 0.267i)7-s + (0.598 + 0.598i)8-s + 0.333i·9-s + (−0.0186 − 0.477i)10-s + 0.690·11-s + (−0.315 + 0.315i)12-s + (0.0863 + 0.0863i)13-s + 0.180i·14-s + (0.576 − 0.0225i)15-s − 0.367·16-s + (−0.202 + 0.202i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.963 - 0.267i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (43, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.963 - 0.267i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.90944 + 0.260183i\)
\(L(\frac12)\)  \(\approx\)  \(1.90944 + 0.260183i\)
\(L(2)\)   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 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 + (-3.39 + 3.67i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good2 \( 1 + (-0.675 + 0.675i)T - 4iT^{2} \)
11 \( 1 - 7.59T + 121T^{2} \)
13 \( 1 + (-1.12 - 1.12i)T + 169iT^{2} \)
17 \( 1 + (3.43 - 3.43i)T - 289iT^{2} \)
19 \( 1 + 26.3iT - 361T^{2} \)
23 \( 1 + (24.2 + 24.2i)T + 529iT^{2} \)
29 \( 1 - 22.3iT - 841T^{2} \)
31 \( 1 + 18.3T + 961T^{2} \)
37 \( 1 + (34.4 - 34.4i)T - 1.36e3iT^{2} \)
41 \( 1 - 37.4T + 1.68e3T^{2} \)
43 \( 1 + (55.1 + 55.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (-40.8 + 40.8i)T - 2.20e3iT^{2} \)
53 \( 1 + (9.39 + 9.39i)T + 2.80e3iT^{2} \)
59 \( 1 - 49.7iT - 3.48e3T^{2} \)
61 \( 1 - 88.3T + 3.72e3T^{2} \)
67 \( 1 + (40.5 - 40.5i)T - 4.48e3iT^{2} \)
71 \( 1 - 136.T + 5.04e3T^{2} \)
73 \( 1 + (-5.85 - 5.85i)T + 5.32e3iT^{2} \)
79 \( 1 - 66.4iT - 6.24e3T^{2} \)
83 \( 1 + (34.8 + 34.8i)T + 6.88e3iT^{2} \)
89 \( 1 + 2.03iT - 7.92e3T^{2} \)
97 \( 1 + (58.4 - 58.4i)T - 9.40e3iT^{2} \)
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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

−13.47016623969460810431573213901, −12.62543839278077829348834992092, −11.70028463183214826510525415212, −10.35855406929580562388784635781, −9.051485342412645457670530279202, −8.483821424593878937681249172969, −6.78789255955853411316682729201, −5.09368617839340368172185134598, −3.89059188715993919794225499585, −2.29913221563469416565161512649, 1.77388266408271202743049270835, 3.77433819928230511262116627695, 5.72570194445150865673831818929, 6.52016089026607719349997592325, 7.64293078726937443585621505790, 9.409251278266254373468500545383, 10.10322795155976706823525379207, 11.30319421366029411702272081001, 12.76048021957725830151903223262, 13.94433626073824245788961924982

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