Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.587 + 0.809i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.79 − 1.79i)2-s + (0.491 + 1.66i)3-s − 4.47i·4-s + (−1.87 + 1.21i)5-s + (3.87 + 2.10i)6-s + (−0.707 − 0.707i)7-s + (−4.45 − 4.45i)8-s + (−2.51 + 1.63i)9-s + (−1.20 + 5.56i)10-s + 1.56i·11-s + (7.43 − 2.19i)12-s + (2.21 − 2.21i)13-s − 2.54·14-s + (−2.93 − 2.52i)15-s − 7.09·16-s + (−3.60 + 3.60i)17-s + ⋯
L(s)  = 1  + (1.27 − 1.27i)2-s + (0.283 + 0.958i)3-s − 2.23i·4-s + (−0.840 + 0.541i)5-s + (1.58 + 0.859i)6-s + (−0.267 − 0.267i)7-s + (−1.57 − 1.57i)8-s + (−0.839 + 0.543i)9-s + (−0.380 + 1.75i)10-s + 0.472i·11-s + (2.14 − 0.634i)12-s + (0.615 − 0.615i)13-s − 0.680·14-s + (−0.757 − 0.652i)15-s − 1.77·16-s + (−0.874 + 0.874i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.587 + 0.809i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (92, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.587 + 0.809i)$
$L(1)$  $\approx$  $1.51480 - 0.772265i$
$L(\frac12)$  $\approx$  $1.51480 - 0.772265i$
$L(\frac{3}{2})$   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 \( 1 + (-0.491 - 1.66i)T \)
5 \( 1 + (1.87 - 1.21i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (-1.79 + 1.79i)T - 2iT^{2} \)
11 \( 1 - 1.56iT - 11T^{2} \)
13 \( 1 + (-2.21 + 2.21i)T - 13iT^{2} \)
17 \( 1 + (3.60 - 3.60i)T - 17iT^{2} \)
19 \( 1 + 1.68iT - 19T^{2} \)
23 \( 1 + (0.995 + 0.995i)T + 23iT^{2} \)
29 \( 1 - 8.91T + 29T^{2} \)
31 \( 1 - 2.74T + 31T^{2} \)
37 \( 1 + (-0.440 - 0.440i)T + 37iT^{2} \)
41 \( 1 + 6.44iT - 41T^{2} \)
43 \( 1 + (5.47 - 5.47i)T - 43iT^{2} \)
47 \( 1 + (3.69 - 3.69i)T - 47iT^{2} \)
53 \( 1 + (2.83 + 2.83i)T + 53iT^{2} \)
59 \( 1 + 5.54T + 59T^{2} \)
61 \( 1 - 7.40T + 61T^{2} \)
67 \( 1 + (3.75 + 3.75i)T + 67iT^{2} \)
71 \( 1 - 3.61iT - 71T^{2} \)
73 \( 1 + (5.89 - 5.89i)T - 73iT^{2} \)
79 \( 1 - 17.0iT - 79T^{2} \)
83 \( 1 + (3.21 + 3.21i)T + 83iT^{2} \)
89 \( 1 - 9.40T + 89T^{2} \)
97 \( 1 + (-4.39 - 4.39i)T + 97iT^{2} \)
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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

−13.57405960146801088000910615120, −12.53065521574232815055942768139, −11.41846511725799346587460356128, −10.68332253335893094846327893578, −9.994103149412449726600665618006, −8.377082999164225975873342972770, −6.36559857743229077679070290163, −4.73920824630201309580699362897, −3.86534869821425061866328240933, −2.79725446955299344494612740399, 3.25865001457220604375637755349, 4.67671495765148965636120027793, 6.12541180591844345699878498504, 6.98647535760392008264507964271, 8.119211132494916816745949650683, 8.836108640022546420578674900858, 11.59581525080306444356239961711, 12.16673080860690164201939417657, 13.27430891491400397731276205638, 13.77180516153882776192860468043

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