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  :  $\chi_{105} (8, \cdot )$
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

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

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