Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.959 - 0.282i$
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 + (1.72 − 0.102i)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 + (2.97 − 0.355i)9-s + (1.28 − 5.30i)10-s + (3.08 + 1.77i)11-s + (6.11 + 3.06i)12-s + (1.28 − 1.28i)13-s + (5.67 − 3.08i)14-s + (0.323 + 3.85i)15-s + (1.85 + 3.21i)16-s + (−0.792 − 2.95i)17-s + ⋯
L(s)  = 1  + (−1.66 − 0.446i)2-s + (0.998 − 0.0593i)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 + (0.992 − 0.118i)9-s + (0.406 − 1.67i)10-s + (0.928 + 0.536i)11-s + (1.76 + 0.884i)12-s + (0.356 − 0.356i)13-s + (1.51 − 0.823i)14-s + (0.0834 + 0.996i)15-s + (0.463 + 0.803i)16-s + (−0.192 − 0.717i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.959 - 0.282i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (32, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.959 - 0.282i)$
$L(1)$  $\approx$  $0.654808 + 0.0942559i$
$L(\frac12)$  $\approx$  $0.654808 + 0.0942559i$
$L(\frac{3}{2})$   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 \( 1 + (-1.72 + 0.102i)T \)
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

−13.91422742785650171137322698058, −12.49578140592207992105511747324, −11.40858744101058397166799426731, −10.20700611484692793655827724922, −9.470671943456809656703646316334, −8.698856052872973090950383450335, −7.42812505936811895415784659701, −6.60093349185068136527334130359, −3.36577378315435895535545116192, −2.18424534059329627715729261420, 1.40907799761246195810057222594, 3.88764638450823581595033456217, 6.29910882516856325165866640480, 7.47313396260684716123764095687, 8.508655490968305911875606659848, 9.222048313882014623761031052898, 9.909477128415759657612519293201, 11.20367231972980237326111410798, 12.82439251279709891260277506697, 13.79525054942047429691038818292

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