Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.584 + 0.811i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.63 − 2.63i)2-s + (2.54 − 1.59i)3-s + 9.88i·4-s + (4.15 − 2.78i)5-s + (−10.8 − 2.49i)6-s + (6.57 − 2.39i)7-s + (15.5 − 15.5i)8-s + (3.91 − 8.10i)9-s + (−18.2 − 3.60i)10-s + 5.66i·11-s + (15.7 + 25.1i)12-s + (1.68 − 1.68i)13-s + (−23.6 − 11.0i)14-s + (6.11 − 13.6i)15-s − 42.1·16-s + (−14.0 + 14.0i)17-s + ⋯
L(s)  = 1  + (−1.31 − 1.31i)2-s + (0.846 − 0.531i)3-s + 2.47i·4-s + (0.830 − 0.556i)5-s + (−1.81 − 0.415i)6-s + (0.939 − 0.342i)7-s + (1.93 − 1.93i)8-s + (0.434 − 0.900i)9-s + (−1.82 − 0.360i)10-s + 0.514i·11-s + (1.31 + 2.09i)12-s + (0.129 − 0.129i)13-s + (−1.68 − 0.786i)14-s + (0.407 − 0.913i)15-s − 2.63·16-s + (−0.824 + 0.824i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.584 + 0.811i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.584 + 0.811i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.499105 - 0.973920i\)
\(L(\frac12)\)  \(\approx\)  \(0.499105 - 0.973920i\)
\(L(2)\)   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 \( 1 + (-2.54 + 1.59i)T \)
5 \( 1 + (-4.15 + 2.78i)T \)
7 \( 1 + (-6.57 + 2.39i)T \)
good2 \( 1 + (2.63 + 2.63i)T + 4iT^{2} \)
11 \( 1 - 5.66iT - 121T^{2} \)
13 \( 1 + (-1.68 + 1.68i)T - 169iT^{2} \)
17 \( 1 + (14.0 - 14.0i)T - 289iT^{2} \)
19 \( 1 + 24.0T + 361T^{2} \)
23 \( 1 + (-3.17 + 3.17i)T - 529iT^{2} \)
29 \( 1 - 24.1T + 841T^{2} \)
31 \( 1 - 23.8iT - 961T^{2} \)
37 \( 1 + (13.8 - 13.8i)T - 1.36e3iT^{2} \)
41 \( 1 + 53.4T + 1.68e3T^{2} \)
43 \( 1 + (-25.9 - 25.9i)T + 1.84e3iT^{2} \)
47 \( 1 + (-27.3 + 27.3i)T - 2.20e3iT^{2} \)
53 \( 1 + (-22.4 + 22.4i)T - 2.80e3iT^{2} \)
59 \( 1 - 14.2iT - 3.48e3T^{2} \)
61 \( 1 - 90.2iT - 3.72e3T^{2} \)
67 \( 1 + (0.492 - 0.492i)T - 4.48e3iT^{2} \)
71 \( 1 - 54.2iT - 5.04e3T^{2} \)
73 \( 1 + (30.4 - 30.4i)T - 5.32e3iT^{2} \)
79 \( 1 - 58.3iT - 6.24e3T^{2} \)
83 \( 1 + (-55.7 - 55.7i)T + 6.88e3iT^{2} \)
89 \( 1 - 109. iT - 7.92e3T^{2} \)
97 \( 1 + (48.0 + 48.0i)T + 9.40e3iT^{2} \)
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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

−12.88555963106463566662353724491, −12.11743834968850010372587331118, −10.73831246737070948765528216829, −9.952910081242455146823159700469, −8.624878410617906739733158792165, −8.430747053504656033453472252998, −6.91496021488073450172941253373, −4.24555712214746688556035672941, −2.34533341253338967305835472565, −1.37930725559908685957762691505, 2.10776973351888749159279285859, 4.93691713575373807543307270669, 6.25770314225606644060699300454, 7.47342262177575161526164011349, 8.608014949617149719098449316034, 9.151798465376365811469023803120, 10.32278618605971614564968790576, 11.05617221654177103774632795451, 13.60970004707241726731879077380, 14.28141891957975957728458713213

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