Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.674 - 0.738i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.50 − 0.869i)2-s + (−1.65 − 2.49i)3-s + (−0.489 − 0.847i)4-s + (−1.93 − 1.11i)5-s + (0.325 + 5.20i)6-s + (2.93 − 6.35i)7-s + 8.65i·8-s + (−3.49 + 8.29i)9-s + (1.94 + 3.36i)10-s + (−11.0 + 6.38i)11-s + (−1.30 + 2.62i)12-s + 0.690·13-s + (−9.93 + 7.01i)14-s + (0.418 + 6.69i)15-s + (5.56 − 9.63i)16-s + (12.5 − 7.22i)17-s + ⋯
L(s)  = 1  + (−0.752 − 0.434i)2-s + (−0.553 − 0.833i)3-s + (−0.122 − 0.211i)4-s + (−0.387 − 0.223i)5-s + (0.0542 + 0.867i)6-s + (0.419 − 0.907i)7-s + 1.08i·8-s + (−0.388 + 0.921i)9-s + (0.194 + 0.336i)10-s + (−1.00 + 0.580i)11-s + (−0.108 + 0.219i)12-s + 0.0530·13-s + (−0.709 + 0.501i)14-s + (0.0279 + 0.446i)15-s + (0.347 − 0.602i)16-s + (0.735 − 0.424i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.674 - 0.738i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (86, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.674 - 0.738i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.100699 + 0.228472i\)
\(L(\frac12)\)  \(\approx\)  \(0.100699 + 0.228472i\)
\(L(2)\)   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 \( 1 + (1.65 + 2.49i)T \)
5 \( 1 + (1.93 + 1.11i)T \)
7 \( 1 + (-2.93 + 6.35i)T \)
good2 \( 1 + (1.50 + 0.869i)T + (2 + 3.46i)T^{2} \)
11 \( 1 + (11.0 - 6.38i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 0.690T + 169T^{2} \)
17 \( 1 + (-12.5 + 7.22i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (13.8 - 23.9i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (37.3 + 21.5i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 26.6iT - 841T^{2} \)
31 \( 1 + (8.94 + 15.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-17.5 + 30.4i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 15.0iT - 1.68e3T^{2} \)
43 \( 1 + 23.1T + 1.84e3T^{2} \)
47 \( 1 + (38.1 + 22.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (78.5 - 45.3i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-21.3 + 12.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (33.0 - 57.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (13.6 + 23.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 76.5iT - 5.04e3T^{2} \)
73 \( 1 + (-24.7 - 42.8i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-18.5 + 32.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 20.0iT - 6.88e3T^{2} \)
89 \( 1 + (-62.7 - 36.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 23.2T + 9.40e3T^{2} \)
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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

−12.65392624496501072923997798723, −11.69058057034368634081958715502, −10.63040645153720850426689775643, −9.974375869975597860904010726483, −8.111458590422587627110033742912, −7.72747522293431053809155896375, −5.95923023660269113785734224539, −4.60101564147969526424409287546, −1.91323524993802085227931602854, −0.24914812453454057573550143046, 3.37409648568818890719972716321, 4.97888841736827729409867100212, 6.29030969510584420444650657345, 7.893373030137760301412346195289, 8.700799252424127342261064941147, 9.808748014978193955445427225397, 10.88276972594052094514713185406, 11.89418542460915147324162831622, 12.95987489508085182976771505675, 14.55143346744060212071918344157

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