Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.871 - 0.490i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.987 + 0.569i)2-s + (2.69 + 1.32i)3-s + (−1.35 − 2.33i)4-s + (1.93 + 1.11i)5-s + (1.90 + 2.83i)6-s + (2.61 + 6.49i)7-s − 7.63i·8-s + (5.50 + 7.11i)9-s + (1.27 + 2.20i)10-s + (12.8 − 7.41i)11-s + (−0.546 − 8.08i)12-s − 23.9·13-s + (−1.12 + 7.89i)14-s + (3.73 + 5.56i)15-s + (−1.04 + 1.81i)16-s + (−5.54 + 3.20i)17-s + ⋯
L(s)  = 1  + (0.493 + 0.284i)2-s + (0.897 + 0.440i)3-s + (−0.337 − 0.584i)4-s + (0.387 + 0.223i)5-s + (0.317 + 0.473i)6-s + (0.372 + 0.927i)7-s − 0.954i·8-s + (0.612 + 0.790i)9-s + (0.127 + 0.220i)10-s + (1.16 − 0.673i)11-s + (−0.0455 − 0.673i)12-s − 1.84·13-s + (−0.0803 + 0.564i)14-s + (0.249 + 0.371i)15-s + (−0.0655 + 0.113i)16-s + (−0.326 + 0.188i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.871 - 0.490i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (86, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.871 - 0.490i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.08495 + 0.546942i\)
\(L(\frac12)\)  \(\approx\)  \(2.08495 + 0.546942i\)
\(L(2)\)   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 \( 1 + (-2.69 - 1.32i)T \)
5 \( 1 + (-1.93 - 1.11i)T \)
7 \( 1 + (-2.61 - 6.49i)T \)
good2 \( 1 + (-0.987 - 0.569i)T + (2 + 3.46i)T^{2} \)
11 \( 1 + (-12.8 + 7.41i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 23.9T + 169T^{2} \)
17 \( 1 + (5.54 - 3.20i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-3.55 + 6.16i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (28.8 + 16.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 32.6iT - 841T^{2} \)
31 \( 1 + (1.00 + 1.73i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-9.06 + 15.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 40.8iT - 1.68e3T^{2} \)
43 \( 1 + 29.7T + 1.84e3T^{2} \)
47 \( 1 + (6.22 + 3.59i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-34.1 + 19.7i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-66.6 + 38.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-13.9 + 24.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-50.9 - 88.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 65.1iT - 5.04e3T^{2} \)
73 \( 1 + (-58.9 - 102. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-17.0 + 29.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 34.5iT - 6.88e3T^{2} \)
89 \( 1 + (-28.7 - 16.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 5.32T + 9.40e3T^{2} \)
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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

−14.05333858503998708545419960492, −12.86620249251843839760644566035, −11.59685663162777430886320168669, −10.00086679200956701640193130685, −9.434745821557850729322982615085, −8.303367723654437796482687953010, −6.66063481086472904693543439740, −5.35215162597001949399756135011, −4.17298150034568791909267615018, −2.32180273531594746186054063297, 2.00892314074813253300250042229, 3.71699790550397075282512545703, 4.79950980704603675215268745392, 7.00204125746175395958318799782, 7.81288941378208584050819813596, 9.118524331029540903191970072197, 10.01118899990990985908845939403, 11.87592148950651692895155211113, 12.44127641401231463461622788923, 13.57051191502863565403699361381

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