Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.164 - 0.986i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.860 + 0.496i)2-s + (2.51 + 1.63i)3-s + (−1.50 + 2.60i)4-s + (1.93 − 1.11i)5-s + (−2.97 − 0.162i)6-s + (−1.66 + 6.79i)7-s − 6.96i·8-s + (3.62 + 8.23i)9-s + (−1.11 + 1.92i)10-s + (−0.568 − 0.328i)11-s + (−8.06 + 4.08i)12-s − 10.1·13-s + (−1.94 − 6.67i)14-s + (6.69 + 0.366i)15-s + (−2.56 − 4.44i)16-s + (16.8 + 9.72i)17-s + ⋯
L(s)  = 1  + (−0.430 + 0.248i)2-s + (0.837 + 0.546i)3-s + (−0.376 + 0.652i)4-s + (0.387 − 0.223i)5-s + (−0.495 − 0.0271i)6-s + (−0.238 + 0.971i)7-s − 0.870i·8-s + (0.402 + 0.915i)9-s + (−0.111 + 0.192i)10-s + (−0.0517 − 0.0298i)11-s + (−0.672 + 0.340i)12-s − 0.781·13-s + (−0.138 − 0.476i)14-s + (0.446 + 0.0244i)15-s + (−0.160 − 0.278i)16-s + (0.990 + 0.571i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.164 - 0.986i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.164 - 0.986i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.830070 + 0.979538i\)
\(L(\frac12)\)  \(\approx\)  \(0.830070 + 0.979538i\)
\(L(2)\)   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 \( 1 + (-2.51 - 1.63i)T \)
5 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 + (1.66 - 6.79i)T \)
good2 \( 1 + (0.860 - 0.496i)T + (2 - 3.46i)T^{2} \)
11 \( 1 + (0.568 + 0.328i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 10.1T + 169T^{2} \)
17 \( 1 + (-16.8 - 9.72i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-9.11 - 15.7i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-3.29 + 1.90i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 50.8iT - 841T^{2} \)
31 \( 1 + (-26.8 + 46.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-7.81 - 13.5i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 57.3iT - 1.68e3T^{2} \)
43 \( 1 - 65.7T + 1.84e3T^{2} \)
47 \( 1 + (22.4 - 12.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-5.64 - 3.25i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (18.7 + 10.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-17.1 - 29.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (39.5 - 68.5i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 39.0iT - 5.04e3T^{2} \)
73 \( 1 + (19.9 - 34.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-31.9 - 55.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 75.6iT - 6.88e3T^{2} \)
89 \( 1 + (-57.0 + 32.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 113.T + 9.40e3T^{2} \)
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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

−13.85126671001522036312843274706, −12.83302425070311325809847797189, −11.94637022639565869289364583231, −9.980118529166501352184932750487, −9.509256334806043224270895330307, −8.398927384983806925561136081336, −7.63526599865736031064205710373, −5.73235754103599653723508853213, −4.14006631906532424445859112750, −2.64363076656771913643208033894, 1.14517892939008724956997259590, 2.99109522879246153084079244297, 4.95709637162747951554725676587, 6.70714055673380045599834810994, 7.72244436821450309716106640683, 9.131212036299928175222681560283, 9.841155159356797628471657556468, 10.80484276029753674676007358353, 12.32772518251022659028359048632, 13.51946188750718654682470183418

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