Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.737 - 0.675i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.59 − 1.59i)2-s + (−1.22 + 1.22i)3-s + 1.11i·4-s + (−1.35 − 4.81i)5-s + 3.91·6-s + (1.87 + 1.87i)7-s + (−4.61 + 4.61i)8-s − 2.99i·9-s + (−5.52 + 9.86i)10-s − 13.7·11-s + (−1.36 − 1.36i)12-s + (−16.4 + 16.4i)13-s − 5.98i·14-s + (7.55 + 4.22i)15-s + 19.2·16-s + (−3.05 − 3.05i)17-s + ⋯
L(s)  = 1  + (−0.799 − 0.799i)2-s + (−0.408 + 0.408i)3-s + 0.278i·4-s + (−0.271 − 0.962i)5-s + 0.652·6-s + (0.267 + 0.267i)7-s + (−0.576 + 0.576i)8-s − 0.333i·9-s + (−0.552 + 0.986i)10-s − 1.24·11-s + (−0.113 − 0.113i)12-s + (−1.26 + 1.26i)13-s − 0.427i·14-s + (0.503 + 0.281i)15-s + 1.20·16-s + (−0.179 − 0.179i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.737 - 0.675i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (22, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.737 - 0.675i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0393813 + 0.101249i\)
\(L(\frac12)\)  \(\approx\)  \(0.0393813 + 0.101249i\)
\(L(2)\)   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 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 + (1.35 + 4.81i)T \)
7 \( 1 + (-1.87 - 1.87i)T \)
good2 \( 1 + (1.59 + 1.59i)T + 4iT^{2} \)
11 \( 1 + 13.7T + 121T^{2} \)
13 \( 1 + (16.4 - 16.4i)T - 169iT^{2} \)
17 \( 1 + (3.05 + 3.05i)T + 289iT^{2} \)
19 \( 1 + 4.66iT - 361T^{2} \)
23 \( 1 + (4.61 - 4.61i)T - 529iT^{2} \)
29 \( 1 + 50.3iT - 841T^{2} \)
31 \( 1 - 11.0T + 961T^{2} \)
37 \( 1 + (44.4 + 44.4i)T + 1.36e3iT^{2} \)
41 \( 1 + 20.5T + 1.68e3T^{2} \)
43 \( 1 + (-41.9 + 41.9i)T - 1.84e3iT^{2} \)
47 \( 1 + (-20.4 - 20.4i)T + 2.20e3iT^{2} \)
53 \( 1 + (46.1 - 46.1i)T - 2.80e3iT^{2} \)
59 \( 1 - 47.2iT - 3.48e3T^{2} \)
61 \( 1 - 33.7T + 3.72e3T^{2} \)
67 \( 1 + (63.1 + 63.1i)T + 4.48e3iT^{2} \)
71 \( 1 + 31.1T + 5.04e3T^{2} \)
73 \( 1 + (19.0 - 19.0i)T - 5.32e3iT^{2} \)
79 \( 1 + 53.2iT - 6.24e3T^{2} \)
83 \( 1 + (-97.3 + 97.3i)T - 6.88e3iT^{2} \)
89 \( 1 - 156. iT - 7.92e3T^{2} \)
97 \( 1 + (76.2 + 76.2i)T + 9.40e3iT^{2} \)
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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

−12.30757270884550102449928535383, −11.78268413270148182190066364822, −10.69031963642121150151644651695, −9.645026080062928268402799486272, −8.909244590889140074946613649685, −7.65151166884749177901602038144, −5.60530471568001401458343600366, −4.55067434857778274451642241889, −2.22717373917717638106180586373, −0.10249653671813863887519817901, 2.98487043885350033638709752303, 5.28763039420818655982997501222, 6.71983967774656425652109004363, 7.57104486451344505654990154641, 8.230814236870805888405007121107, 10.04705504416025854901409146516, 10.67701753584954965825141454331, 12.13027958108330228553622342829, 13.01801601950036989563411762694, 14.48771587236868105340013900133

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