Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.472 + 0.881i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.644 + 0.372i)2-s + (0.164 − 2.99i)3-s + (−1.72 − 2.98i)4-s + (−1.93 − 1.11i)5-s + (1.22 − 1.87i)6-s + (−5.98 + 3.63i)7-s − 5.54i·8-s + (−8.94 − 0.983i)9-s + (−0.832 − 1.44i)10-s + (8.35 − 4.82i)11-s + (−9.22 + 4.67i)12-s − 0.0661·13-s + (−5.21 + 0.116i)14-s + (−3.66 + 5.61i)15-s + (−4.82 + 8.36i)16-s + (28.1 − 16.2i)17-s + ⋯
L(s)  = 1  + (0.322 + 0.186i)2-s + (0.0547 − 0.998i)3-s + (−0.430 − 0.746i)4-s + (−0.387 − 0.223i)5-s + (0.203 − 0.311i)6-s + (−0.854 + 0.519i)7-s − 0.692i·8-s + (−0.994 − 0.109i)9-s + (−0.0832 − 0.144i)10-s + (0.759 − 0.438i)11-s + (−0.768 + 0.389i)12-s − 0.00508·13-s + (−0.372 + 0.00832i)14-s + (−0.244 + 0.374i)15-s + (−0.301 + 0.522i)16-s + (1.65 − 0.954i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.472 + 0.881i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (86, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.472 + 0.881i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.590890 - 0.987799i\)
\(L(\frac12)\)  \(\approx\)  \(0.590890 - 0.987799i\)
\(L(2)\)   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 \( 1 + (-0.164 + 2.99i)T \)
5 \( 1 + (1.93 + 1.11i)T \)
7 \( 1 + (5.98 - 3.63i)T \)
good2 \( 1 + (-0.644 - 0.372i)T + (2 + 3.46i)T^{2} \)
11 \( 1 + (-8.35 + 4.82i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 0.0661T + 169T^{2} \)
17 \( 1 + (-28.1 + 16.2i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-12.7 + 22.1i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-8.49 - 4.90i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 6.58iT - 841T^{2} \)
31 \( 1 + (16.4 + 28.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (27.3 - 47.3i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 14.8iT - 1.68e3T^{2} \)
43 \( 1 + 14.3T + 1.84e3T^{2} \)
47 \( 1 + (-63.7 - 36.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (44.3 - 25.5i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-75.2 + 43.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (12.5 - 21.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-24.0 - 41.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 113. iT - 5.04e3T^{2} \)
73 \( 1 + (21.2 + 36.7i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-9.49 + 16.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 46.8iT - 6.88e3T^{2} \)
89 \( 1 + (45.4 + 26.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 147.T + 9.40e3T^{2} \)
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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

−13.31988958826294070901427305742, −12.27545611824442893102682152913, −11.42890068709175851171320978709, −9.684840408899337809784821579258, −8.902005520954300791664956944910, −7.37926448883084141045068592050, −6.26608492929076165812370701209, −5.24320176209637830965475785351, −3.23991013274241250025739166007, −0.830408026951502604632650951768, 3.41985287127270829262578627248, 3.91161628893751355963495991985, 5.53146634370585441013645498866, 7.30013977037292966526102586983, 8.546298907267600271666148244100, 9.695794021958082343753207555846, 10.58903322537047552701570299916, 11.97307952781481764209079472915, 12.61871621477088564441535179117, 14.11419614712382242132091559418

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