Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.972 - 0.231i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.60 − 2.77i)2-s + (0.199 + 2.99i)3-s + (−3.15 + 5.45i)4-s + (1.12 − 4.87i)5-s + (7.99 − 5.35i)6-s + (−6.02 − 3.55i)7-s + 7.38·8-s + (−8.92 + 1.19i)9-s + (−15.3 + 4.68i)10-s + (−10.5 − 6.09i)11-s + (−16.9 − 8.34i)12-s − 8.47i·13-s + (−0.220 + 22.4i)14-s + (14.8 + 2.39i)15-s + (0.744 + 1.29i)16-s + (−5.29 + 9.17i)17-s + ⋯
L(s)  = 1  + (−0.802 − 1.38i)2-s + (0.0666 + 0.997i)3-s + (−0.787 + 1.36i)4-s + (0.225 − 0.974i)5-s + (1.33 − 0.893i)6-s + (−0.861 − 0.508i)7-s + 0.923·8-s + (−0.991 + 0.132i)9-s + (−1.53 + 0.468i)10-s + (−0.959 − 0.553i)11-s + (−1.41 − 0.695i)12-s − 0.651i·13-s + (−0.0157 + 1.60i)14-s + (0.987 + 0.159i)15-s + (0.0465 + 0.0806i)16-s + (−0.311 + 0.539i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.972 - 0.231i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (74, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.972 - 0.231i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0473046 + 0.402687i\)
\(L(\frac12)\)  \(\approx\)  \(0.0473046 + 0.402687i\)
\(L(2)\)   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 \( 1 + (-0.199 - 2.99i)T \)
5 \( 1 + (-1.12 + 4.87i)T \)
7 \( 1 + (6.02 + 3.55i)T \)
good2 \( 1 + (1.60 + 2.77i)T + (-2 + 3.46i)T^{2} \)
11 \( 1 + (10.5 + 6.09i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 8.47iT - 169T^{2} \)
17 \( 1 + (5.29 - 9.17i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (10.0 + 17.4i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-15.2 - 26.4i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 42.8iT - 841T^{2} \)
31 \( 1 + (6.11 - 10.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-28.8 + 16.6i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 6.40iT - 1.68e3T^{2} \)
43 \( 1 - 20.0iT - 1.84e3T^{2} \)
47 \( 1 + (-11.8 - 20.5i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-43.9 + 76.1i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-41.9 - 24.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-10.4 - 18.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-19.6 - 11.3i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 2.44iT - 5.04e3T^{2} \)
73 \( 1 + (76.5 + 44.2i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (3.20 + 5.54i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 103.T + 6.88e3T^{2} \)
89 \( 1 + (-54.2 + 31.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 140. iT - 9.40e3T^{2} \)
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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

−12.98706292285323549180345538562, −11.53354607228341996191714095369, −10.63113156594064376264726542647, −9.857645134541959960951601828030, −9.055101259416567044687545310515, −8.112069098850816843602449275615, −5.66905833840923051791401711936, −4.09204109340214056835005735534, −2.78253620409727807100523619875, −0.36598620097610606500185232684, 2.62076718196602492571026820581, 5.60931303625732314735121613274, 6.64931391826308475258410475308, 7.16470617438374535505945858961, 8.376332373588181368209500000668, 9.431323380835068332259586792863, 10.60672319549457365100220100422, 12.18903939473629991628440541840, 13.26504185449689888375113986307, 14.42535463241770366658912452701

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