Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.770 + 0.637i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.35 − 0.631i)2-s + (−1.54 − 0.775i)3-s + (3.42 − 1.97i)4-s + (−0.0540 + 2.23i)5-s + (−4.13 − 0.849i)6-s + (−1.91 − 1.82i)7-s + (3.36 − 3.36i)8-s + (1.79 + 2.40i)9-s + (1.28 + 5.30i)10-s + (−3.08 + 1.77i)11-s + (−6.83 + 0.406i)12-s + (1.28 + 1.28i)13-s + (−5.67 − 3.08i)14-s + (1.81 − 3.42i)15-s + (1.85 − 3.21i)16-s + (0.792 − 2.95i)17-s + ⋯
L(s)  = 1  + (1.66 − 0.446i)2-s + (−0.894 − 0.447i)3-s + (1.71 − 0.987i)4-s + (−0.0241 + 0.999i)5-s + (−1.68 − 0.346i)6-s + (−0.725 − 0.688i)7-s + (1.19 − 1.19i)8-s + (0.599 + 0.800i)9-s + (0.406 + 1.67i)10-s + (−0.928 + 0.536i)11-s + (−1.97 + 0.117i)12-s + (0.356 + 0.356i)13-s + (−1.51 − 0.823i)14-s + (0.469 − 0.883i)15-s + (0.463 − 0.803i)16-s + (0.192 − 0.717i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.770 + 0.637i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (23, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.770 + 0.637i)$
$L(1)$  $\approx$  $1.62806 - 0.585717i$
$L(\frac12)$  $\approx$  $1.62806 - 0.585717i$
$L(\frac{3}{2})$   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 \( 1 + (1.54 + 0.775i)T \)
5 \( 1 + (0.0540 - 2.23i)T \)
7 \( 1 + (1.91 + 1.82i)T \)
good2 \( 1 + (-2.35 + 0.631i)T + (1.73 - i)T^{2} \)
11 \( 1 + (3.08 - 1.77i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.28 - 1.28i)T + 13iT^{2} \)
17 \( 1 + (-0.792 + 2.95i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.331 - 0.191i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.658 + 2.45i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.51T + 29T^{2} \)
31 \( 1 + (-0.323 - 0.561i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.34 + 5.00i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 10.1iT - 41T^{2} \)
43 \( 1 + (0.335 + 0.335i)T + 43iT^{2} \)
47 \( 1 + (-2.80 + 0.751i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.04 + 0.815i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.81 - 6.60i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.45 - 9.45i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.3 - 3.31i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.06iT - 71T^{2} \)
73 \( 1 + (0.849 - 3.17i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.21 + 1.85i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.973 - 0.973i)T - 83iT^{2} \)
89 \( 1 + (-1.51 + 2.63i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.3 - 10.3i)T - 97iT^{2} \)
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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

−13.58056449639412908991287292703, −12.66388240440585691984774421719, −11.82572555280642318453772648011, −10.76168491432123600880255555463, −10.21634446088090384657214377407, −7.31294601876405843772222570020, −6.60236395183664917839902528809, −5.47937492612628641895201661495, −4.11073559390851045737695272659, −2.57840070936464811318725115299, 3.35825673050670158052135747612, 4.77446497453703299763703119885, 5.63148947963667615192455781611, 6.38156973014447193831862892205, 8.180577503758516316971543172110, 9.780680426967857996457038678964, 11.20935306810921370871956843532, 12.25166257730708830343215997665, 12.80601086764414156635374259021, 13.60261886039178459235992854933

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