Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.953 - 0.302i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.897 − 1.55i)2-s + (−2.91 + 0.725i)3-s + (0.387 + 0.671i)4-s + (3.31 + 3.74i)5-s + (−1.48 + 5.17i)6-s + (−1.39 + 6.85i)7-s + 8.57·8-s + (7.94 − 4.22i)9-s + (8.79 − 1.78i)10-s + (−1.00 + 0.581i)11-s + (−1.61 − 1.67i)12-s − 12.4i·13-s + (9.41 + 8.32i)14-s + (−12.3 − 8.49i)15-s + (6.14 − 10.6i)16-s + (9.31 + 16.1i)17-s + ⋯
L(s)  = 1  + (0.448 − 0.777i)2-s + (−0.970 + 0.241i)3-s + (0.0969 + 0.167i)4-s + (0.662 + 0.748i)5-s + (−0.247 + 0.862i)6-s + (−0.199 + 0.979i)7-s + 1.07·8-s + (0.883 − 0.469i)9-s + (0.879 − 0.178i)10-s + (−0.0916 + 0.0528i)11-s + (−0.134 − 0.139i)12-s − 0.959i·13-s + (0.672 + 0.594i)14-s + (−0.824 − 0.566i)15-s + (0.384 − 0.665i)16-s + (0.547 + 0.949i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.953 - 0.302i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (44, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.953 - 0.302i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.47459 + 0.228537i\)
\(L(\frac12)\)  \(\approx\)  \(1.47459 + 0.228537i\)
\(L(2)\)   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 \( 1 + (2.91 - 0.725i)T \)
5 \( 1 + (-3.31 - 3.74i)T \)
7 \( 1 + (1.39 - 6.85i)T \)
good2 \( 1 + (-0.897 + 1.55i)T + (-2 - 3.46i)T^{2} \)
11 \( 1 + (1.00 - 0.581i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 12.4iT - 169T^{2} \)
17 \( 1 + (-9.31 - 16.1i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (15.2 - 26.3i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-13.7 + 23.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 52.6iT - 841T^{2} \)
31 \( 1 + (17.2 + 29.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-0.357 - 0.206i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 17.2iT - 1.68e3T^{2} \)
43 \( 1 - 7.86iT - 1.84e3T^{2} \)
47 \( 1 + (-17.4 + 30.1i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (17.8 + 30.9i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-32.3 + 18.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-25.4 + 44.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-24.9 + 14.4i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 66.8iT - 5.04e3T^{2} \)
73 \( 1 + (-46.7 + 27.0i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (16.6 - 28.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 72.0T + 6.88e3T^{2} \)
89 \( 1 + (-41.4 - 23.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 66.7iT - 9.40e3T^{2} \)
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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

−13.01418306407309469390047649517, −12.54475690631010337202131508852, −11.50345266648139944959470090499, −10.53635514307490227854665542719, −9.915738942365409339510515021583, −8.007937786821701502611548546870, −6.38780192583831413078696661474, −5.50646854727579582746101589908, −3.77234684933574703523074474376, −2.21643567473004206598140787716, 1.27745791688507778411513181386, 4.58821510741569825762490029431, 5.34887480692151686938557320669, 6.67370082564247757824498020038, 7.27741105538359785776157284541, 9.223194184606070836345323800595, 10.42902956400630642480667644428, 11.31447666967581857997029463902, 12.75856626203735151549590216616, 13.52110944456932455560070812235

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