Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.956 - 0.290i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.50·2-s + 1.73i·3-s + 8.27·4-s − 2.23i·5-s + 6.06i·6-s + (−6.69 + 2.03i)7-s + 14.9·8-s − 2.99·9-s − 7.83i·10-s − 2.03·11-s + 14.3i·12-s − 18.0i·13-s + (−23.4 + 7.13i)14-s + 3.87·15-s + 19.3·16-s + 1.07i·17-s + ⋯
L(s)  = 1  + 1.75·2-s + 0.577i·3-s + 2.06·4-s − 0.447i·5-s + 1.01i·6-s + (−0.956 + 0.290i)7-s + 1.87·8-s − 0.333·9-s − 0.783i·10-s − 0.184·11-s + 1.19i·12-s − 1.38i·13-s + (−1.67 + 0.509i)14-s + 0.258·15-s + 1.21·16-s + 0.0631i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.956 - 0.290i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (76, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.956 - 0.290i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(3.02488 + 0.449624i\)
\(L(\frac12)\)  \(\approx\)  \(3.02488 + 0.449624i\)
\(L(2)\)   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 \( 1 - 1.73iT \)
5 \( 1 + 2.23iT \)
7 \( 1 + (6.69 - 2.03i)T \)
good2 \( 1 - 3.50T + 4T^{2} \)
11 \( 1 + 2.03T + 121T^{2} \)
13 \( 1 + 18.0iT - 169T^{2} \)
17 \( 1 - 1.07iT - 289T^{2} \)
19 \( 1 - 28.7iT - 361T^{2} \)
23 \( 1 - 24.8T + 529T^{2} \)
29 \( 1 + 38.4T + 841T^{2} \)
31 \( 1 + 44.0iT - 961T^{2} \)
37 \( 1 - 37.2T + 1.36e3T^{2} \)
41 \( 1 - 49.9iT - 1.68e3T^{2} \)
43 \( 1 + 9.58T + 1.84e3T^{2} \)
47 \( 1 - 55.6iT - 2.20e3T^{2} \)
53 \( 1 - 57.4T + 2.80e3T^{2} \)
59 \( 1 - 101. iT - 3.48e3T^{2} \)
61 \( 1 + 31.9iT - 3.72e3T^{2} \)
67 \( 1 - 95.7T + 4.48e3T^{2} \)
71 \( 1 + 25.8T + 5.04e3T^{2} \)
73 \( 1 + 95.6iT - 5.32e3T^{2} \)
79 \( 1 - 28.1T + 6.24e3T^{2} \)
83 \( 1 + 103. iT - 6.88e3T^{2} \)
89 \( 1 - 29.3iT - 7.92e3T^{2} \)
97 \( 1 + 67.6iT - 9.40e3T^{2} \)
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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

−13.22620749833861708032850748650, −12.91491561907220389357491343904, −11.85389232128567032967592811420, −10.67037686690589853454261874385, −9.500541686157581808648010977118, −7.78353005969709190664887274379, −6.08309355905801837521761843714, −5.40609977319768783533801597773, −3.99158677212916992690628271758, −2.89988737057636046266227934663, 2.50629075228292060907591811509, 3.78747182477589951776794821808, 5.27254538610868259803939914175, 6.78382406646877549013185762351, 6.96259081073287622452861910324, 9.192224632605330162802211565691, 10.88204949766889923834702905603, 11.68576554982934009329345627547, 12.79275707997601444624406076397, 13.42313435107440278352923778511

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