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  :  $\chi_{105} (23, \cdot )$
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

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

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