Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.860 + 0.509i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.46 − 0.391i)2-s + (1.49 + 0.879i)3-s + (0.246 + 0.142i)4-s + (−0.207 − 2.22i)5-s + (−1.83 − 1.86i)6-s + (2.36 − 1.17i)7-s + (1.83 + 1.83i)8-s + (1.45 + 2.62i)9-s + (−0.567 + 3.33i)10-s + (−0.791 − 0.457i)11-s + (0.243 + 0.429i)12-s + (3.07 − 3.07i)13-s + (−3.92 + 0.791i)14-s + (1.64 − 3.50i)15-s + (−2.24 − 3.88i)16-s + (0.311 + 1.16i)17-s + ⋯
L(s)  = 1  + (−1.03 − 0.276i)2-s + (0.861 + 0.507i)3-s + (0.123 + 0.0712i)4-s + (−0.0929 − 0.995i)5-s + (−0.749 − 0.762i)6-s + (0.895 − 0.444i)7-s + (0.648 + 0.648i)8-s + (0.484 + 0.874i)9-s + (−0.179 + 1.05i)10-s + (−0.238 − 0.137i)11-s + (0.0702 + 0.124i)12-s + (0.854 − 0.854i)13-s + (−1.04 + 0.211i)14-s + (0.425 − 0.905i)15-s + (−0.561 − 0.971i)16-s + (0.0755 + 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.860 + 0.509i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (32, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.860 + 0.509i)$
$L(1)$  $\approx$  $0.778710 - 0.213469i$
$L(\frac12)$  $\approx$  $0.778710 - 0.213469i$
$L(\frac{3}{2})$   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 \( 1 + (-1.49 - 0.879i)T \)
5 \( 1 + (0.207 + 2.22i)T \)
7 \( 1 + (-2.36 + 1.17i)T \)
good2 \( 1 + (1.46 + 0.391i)T + (1.73 + i)T^{2} \)
11 \( 1 + (0.791 + 0.457i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.07 + 3.07i)T - 13iT^{2} \)
17 \( 1 + (-0.311 - 1.16i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (5.95 - 3.43i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.505 - 1.88i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 2.72T + 29T^{2} \)
31 \( 1 + (2.31 - 4.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.207 + 0.774i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.922iT - 41T^{2} \)
43 \( 1 + (4.80 - 4.80i)T - 43iT^{2} \)
47 \( 1 + (-10.1 - 2.71i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (10.6 - 2.85i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.94 + 8.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.533 - 0.924i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.83 + 1.83i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 0.557iT - 71T^{2} \)
73 \( 1 + (0.564 + 2.10i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.62 - 1.51i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.38 - 2.38i)T + 83iT^{2} \)
89 \( 1 + (-5.64 - 9.78i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.58 + 1.58i)T + 97iT^{2} \)
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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

−13.72193250691506391552183513742, −12.77209802507933055247881066000, −11.04249291441310211086197178559, −10.37744975470307876453951565138, −9.213576198273443383511374104965, −8.271531537973568583866600883295, −7.925622455267189200437780252487, −5.27962775951509313949418947331, −4.04405839665633709146452292090, −1.62700938985477079762534734678, 2.12400642151089781795677030840, 4.07426869154700117392793165439, 6.51215534782253831858188002883, 7.46806916712597404758965733486, 8.437453028826817647129067522572, 9.176179365354942359036782287098, 10.49464886770197310644675178294, 11.54224342120299500546981494748, 13.03111896231221299551573921741, 13.97868878897746856923940573401

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