Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.935 - 0.353i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.67 − 1.67i)2-s + (−1.56 + 2.55i)3-s − 1.58i·4-s + (0.529 + 4.97i)5-s + (1.66 + 6.89i)6-s + (6.78 − 1.73i)7-s + (4.03 + 4.03i)8-s + (−4.10 − 8.01i)9-s + (9.19 + 7.42i)10-s + 4.41i·11-s + (4.06 + 2.48i)12-s + (1.62 + 1.62i)13-s + (8.43 − 14.2i)14-s + (−13.5 − 6.42i)15-s + 19.8·16-s + (−13.9 − 13.9i)17-s + ⋯
L(s)  = 1  + (0.835 − 0.835i)2-s + (−0.521 + 0.853i)3-s − 0.397i·4-s + (0.105 + 0.994i)5-s + (0.277 + 1.14i)6-s + (0.968 − 0.247i)7-s + (0.503 + 0.503i)8-s + (−0.455 − 0.890i)9-s + (0.919 + 0.742i)10-s + 0.401i·11-s + (0.338 + 0.207i)12-s + (0.124 + 0.124i)13-s + (0.602 − 1.01i)14-s + (−0.903 − 0.428i)15-s + 1.23·16-s + (−0.819 − 0.819i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.935 - 0.353i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (62, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.935 - 0.353i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.84532 + 0.337128i\)
\(L(\frac12)\)  \(\approx\)  \(1.84532 + 0.337128i\)
\(L(2)\)   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 \( 1 + (1.56 - 2.55i)T \)
5 \( 1 + (-0.529 - 4.97i)T \)
7 \( 1 + (-6.78 + 1.73i)T \)
good2 \( 1 + (-1.67 + 1.67i)T - 4iT^{2} \)
11 \( 1 - 4.41iT - 121T^{2} \)
13 \( 1 + (-1.62 - 1.62i)T + 169iT^{2} \)
17 \( 1 + (13.9 + 13.9i)T + 289iT^{2} \)
19 \( 1 + 0.694T + 361T^{2} \)
23 \( 1 + (23.1 + 23.1i)T + 529iT^{2} \)
29 \( 1 - 49.1T + 841T^{2} \)
31 \( 1 + 33.8iT - 961T^{2} \)
37 \( 1 + (-2.02 - 2.02i)T + 1.36e3iT^{2} \)
41 \( 1 - 32.5T + 1.68e3T^{2} \)
43 \( 1 + (30.4 - 30.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (18.7 + 18.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (33.8 + 33.8i)T + 2.80e3iT^{2} \)
59 \( 1 + 23.1iT - 3.48e3T^{2} \)
61 \( 1 - 12.9iT - 3.72e3T^{2} \)
67 \( 1 + (56.3 + 56.3i)T + 4.48e3iT^{2} \)
71 \( 1 + 92.7iT - 5.04e3T^{2} \)
73 \( 1 + (-95.4 - 95.4i)T + 5.32e3iT^{2} \)
79 \( 1 - 100. iT - 6.24e3T^{2} \)
83 \( 1 + (-5.62 + 5.62i)T - 6.88e3iT^{2} \)
89 \( 1 - 158. iT - 7.92e3T^{2} \)
97 \( 1 + (37.1 - 37.1i)T - 9.40e3iT^{2} \)
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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

−13.73576619803063591846869225789, −12.21375175214651482363152541033, −11.38965987226757882890989373912, −10.79745985460534712793587322568, −9.845048913521335069840096346511, −8.097273200089719597171510574648, −6.49874662273754251975539691515, −4.92450209435073703139195342403, −4.05647938009062051802581766029, −2.50464403759182809581020598010, 1.46937766676904354976203739990, 4.48937429597419750367048833137, 5.46017841559173339397842820466, 6.33958568192421526639674381710, 7.75891484916194769642514942701, 8.607559204015342872479168001424, 10.50507364578969081015012323618, 11.77321744406308502013401763805, 12.64762422805875175796743592507, 13.58386648383985942782945000834

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