Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.0726 + 0.997i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.56 + 0.955i)2-s + (−1.67 − 0.448i)3-s + (8.33 − 4.81i)4-s + (−1.05 + 4.88i)5-s + 6.39·6-s + (−6.28 + 3.07i)7-s + (−14.6 + 14.6i)8-s + (2.59 + 1.50i)9-s + (−0.890 − 18.4i)10-s + (−4.09 − 7.09i)11-s + (−16.0 + 4.31i)12-s + (14.0 − 14.0i)13-s + (19.4 − 16.9i)14-s + (3.96 − 7.70i)15-s + (19.0 − 32.9i)16-s + (1.81 − 6.76i)17-s + ⋯
L(s)  = 1  + (−1.78 + 0.477i)2-s + (−0.557 − 0.149i)3-s + (2.08 − 1.20i)4-s + (−0.211 + 0.977i)5-s + 1.06·6-s + (−0.898 + 0.439i)7-s + (−1.83 + 1.83i)8-s + (0.288 + 0.166i)9-s + (−0.0890 − 1.84i)10-s + (−0.372 − 0.644i)11-s + (−1.34 + 0.359i)12-s + (1.08 − 1.08i)13-s + (1.39 − 1.21i)14-s + (0.264 − 0.513i)15-s + (1.18 − 2.06i)16-s + (0.106 − 0.397i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0726 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0726 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.0726 + 0.997i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.0726 + 0.997i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.144324 - 0.134190i\)
\(L(\frac12)\)  \(\approx\)  \(0.144324 - 0.134190i\)
\(L(2)\)   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 \( 1 + (1.67 + 0.448i)T \)
5 \( 1 + (1.05 - 4.88i)T \)
7 \( 1 + (6.28 - 3.07i)T \)
good2 \( 1 + (3.56 - 0.955i)T + (3.46 - 2i)T^{2} \)
11 \( 1 + (4.09 + 7.09i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-14.0 + 14.0i)T - 169iT^{2} \)
17 \( 1 + (-1.81 + 6.76i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (18.2 + 10.5i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (8.43 + 31.4i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + 22.1iT - 841T^{2} \)
31 \( 1 + (-13.7 - 23.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-14.9 + 4.01i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 0.496T + 1.68e3T^{2} \)
43 \( 1 + (33.4 - 33.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (-24.1 + 6.46i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (34.3 + 9.20i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (15.5 - 9.00i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-13.4 + 23.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (1.44 - 5.37i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 105.T + 5.04e3T^{2} \)
73 \( 1 + (81.6 + 21.8i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (86.5 + 49.9i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-95.6 + 95.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (-54.3 - 31.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-59.0 - 59.0i)T + 9.40e3iT^{2} \)
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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

−13.18159020530643279897509084454, −11.69813023983345558229164409923, −10.66637088196189260670219556168, −10.23718083703956371468836032372, −8.802730331955009912946715007711, −7.85358453253258254190026015584, −6.53114742927717602894968991547, −6.04943437927188563517964260364, −2.79260735813472857357774783054, −0.27995665422362412495114019522, 1.50962308950228602743294467948, 3.93666689331913444543272244773, 6.19354946535361654807561337130, 7.42884370281941824229517705453, 8.625155292851558274619942008952, 9.529458093011853806384557847356, 10.35108790646142807420057023831, 11.42523024650146404582987867347, 12.31715370882094674873212086201, 13.28544632418972481687585166997

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