Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.593 - 0.804i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.79·2-s + 1.73i·3-s + 3.79·4-s − 2.23i·5-s − 4.83i·6-s + (4.15 + 5.63i)7-s + 0.578·8-s − 2.99·9-s + 6.24i·10-s − 18.9·11-s + 6.56i·12-s + 10.9i·13-s + (−11.6 − 15.7i)14-s + 3.87·15-s − 16.7·16-s + 22.3i·17-s + ⋯
L(s)  = 1  − 1.39·2-s + 0.577i·3-s + 0.948·4-s − 0.447i·5-s − 0.805i·6-s + (0.593 + 0.804i)7-s + 0.0723·8-s − 0.333·9-s + 0.624i·10-s − 1.72·11-s + 0.547i·12-s + 0.844i·13-s + (−0.829 − 1.12i)14-s + 0.258·15-s − 1.04·16-s + 1.31i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.593 - 0.804i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.593 - 0.804i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (76, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.593 - 0.804i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.202606 + 0.401437i\)
\(L(\frac12)\)  \(\approx\)  \(0.202606 + 0.401437i\)
\(L(2)\)   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 \( 1 + (-4.15 - 5.63i)T \)
good2 \( 1 + 2.79T + 4T^{2} \)
11 \( 1 + 18.9T + 121T^{2} \)
13 \( 1 - 10.9iT - 169T^{2} \)
17 \( 1 - 22.3iT - 289T^{2} \)
19 \( 1 - 19.6iT - 361T^{2} \)
23 \( 1 + 31.9T + 529T^{2} \)
29 \( 1 - 39.9T + 841T^{2} \)
31 \( 1 + 36.6iT - 961T^{2} \)
37 \( 1 - 8.94T + 1.36e3T^{2} \)
41 \( 1 - 37.6iT - 1.68e3T^{2} \)
43 \( 1 + 18.8T + 1.84e3T^{2} \)
47 \( 1 - 49.3iT - 2.20e3T^{2} \)
53 \( 1 - 49.2T + 2.80e3T^{2} \)
59 \( 1 + 35.2iT - 3.48e3T^{2} \)
61 \( 1 + 63.4iT - 3.72e3T^{2} \)
67 \( 1 - 21.3T + 4.48e3T^{2} \)
71 \( 1 - 36.2T + 5.04e3T^{2} \)
73 \( 1 - 6.66iT - 5.32e3T^{2} \)
79 \( 1 + 16.2T + 6.24e3T^{2} \)
83 \( 1 - 36.7iT - 6.88e3T^{2} \)
89 \( 1 - 88.0iT - 7.92e3T^{2} \)
97 \( 1 + 133. iT - 9.40e3T^{2} \)
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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

−14.01841470292065194139303389041, −12.62082075552412121263511790913, −11.41999917964680342122604275450, −10.36990252454017672188159090718, −9.634855013615896950451006728853, −8.256411030146145078230295384896, −8.092952931459838735743323723077, −5.93908391441143012209557353923, −4.52330250120004627104984218749, −2.06014887344532381242191703502, 0.51063846384969029906319308001, 2.54400904959764824688823624802, 5.06005201179218654675815240708, 7.01750396474060923988551921446, 7.70711336064291108466150605166, 8.544408372490127519899916334328, 10.17914488186298381058474710026, 10.59703106163217013093933193170, 11.76900761146268282593532108926, 13.33259629105202563913402623902

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