Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.357 + 0.934i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.01 − 1.01i)2-s + (1.22 − 1.22i)3-s − 1.94i·4-s + (3.97 − 3.03i)5-s − 2.48·6-s + (1.87 + 1.87i)7-s + (−6.02 + 6.02i)8-s − 2.99i·9-s + (−7.09 − 0.953i)10-s − 10.7·11-s + (−2.38 − 2.38i)12-s + (15.5 − 15.5i)13-s − 3.78i·14-s + (1.15 − 8.58i)15-s + 4.41·16-s + (−13.8 − 13.8i)17-s + ⋯
L(s)  = 1  + (−0.506 − 0.506i)2-s + (0.408 − 0.408i)3-s − 0.487i·4-s + (0.794 − 0.606i)5-s − 0.413·6-s + (0.267 + 0.267i)7-s + (−0.753 + 0.753i)8-s − 0.333i·9-s + (−0.709 − 0.0953i)10-s − 0.973·11-s + (−0.198 − 0.198i)12-s + (1.19 − 1.19i)13-s − 0.270i·14-s + (0.0768 − 0.572i)15-s + 0.275·16-s + (−0.816 − 0.816i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 + 0.934i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.357 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.357 + 0.934i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (22, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.357 + 0.934i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.717042 - 1.04192i\)
\(L(\frac12)\)  \(\approx\)  \(0.717042 - 1.04192i\)
\(L(2)\)   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 \( 1 + (-3.97 + 3.03i)T \)
7 \( 1 + (-1.87 - 1.87i)T \)
good2 \( 1 + (1.01 + 1.01i)T + 4iT^{2} \)
11 \( 1 + 10.7T + 121T^{2} \)
13 \( 1 + (-15.5 + 15.5i)T - 169iT^{2} \)
17 \( 1 + (13.8 + 13.8i)T + 289iT^{2} \)
19 \( 1 - 33.1iT - 361T^{2} \)
23 \( 1 + (-8.39 + 8.39i)T - 529iT^{2} \)
29 \( 1 - 42.0iT - 841T^{2} \)
31 \( 1 - 44.7T + 961T^{2} \)
37 \( 1 + (-21.6 - 21.6i)T + 1.36e3iT^{2} \)
41 \( 1 + 22.6T + 1.68e3T^{2} \)
43 \( 1 + (-9.41 + 9.41i)T - 1.84e3iT^{2} \)
47 \( 1 + (-46.0 - 46.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (-51.4 + 51.4i)T - 2.80e3iT^{2} \)
59 \( 1 + 5.55iT - 3.48e3T^{2} \)
61 \( 1 + 49.2T + 3.72e3T^{2} \)
67 \( 1 + (38.8 + 38.8i)T + 4.48e3iT^{2} \)
71 \( 1 - 23.5T + 5.04e3T^{2} \)
73 \( 1 + (60.1 - 60.1i)T - 5.32e3iT^{2} \)
79 \( 1 + 8.04iT - 6.24e3T^{2} \)
83 \( 1 + (5.76 - 5.76i)T - 6.88e3iT^{2} \)
89 \( 1 - 62.2iT - 7.92e3T^{2} \)
97 \( 1 + (-21.9 - 21.9i)T + 9.40e3iT^{2} \)
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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

−13.21444429569522309338613465994, −12.21147022705578314173999537839, −10.80087061230646201005671712444, −10.04868425684991464342398230004, −8.831263887569637731790553049913, −8.141539380765861750442813767814, −6.16213448371115045493357819958, −5.19459845650228502326731994476, −2.68791798712816473359462782124, −1.20092475946463513272744461365, 2.60175289541509538553588069582, 4.25376200906677366995727186218, 6.18774010941217512457482886476, 7.22572080491040419019518166261, 8.516892327175528067339633706451, 9.286709877777656911093724668842, 10.52674302939643162525610836752, 11.47712156376904132378847558992, 13.42173353103811602386977724475, 13.52469108394550638072046519305

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