Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.890 - 0.454i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.907 + 0.243i)2-s + (1.12 + 1.31i)3-s + (−0.967 − 0.558i)4-s + (2.12 − 0.695i)5-s + (0.700 + 1.46i)6-s + (−2.64 + 0.0144i)7-s + (−2.07 − 2.07i)8-s + (−0.470 + 2.96i)9-s + (2.09 − 0.114i)10-s + (0.630 + 0.363i)11-s + (−0.352 − 1.90i)12-s + (−1.44 + 1.44i)13-s + (−2.40 − 0.630i)14-s + (3.30 + 2.01i)15-s + (−0.257 − 0.446i)16-s + (−1.90 − 7.09i)17-s + ⋯
L(s)  = 1  + (0.641 + 0.171i)2-s + (0.649 + 0.760i)3-s + (−0.483 − 0.279i)4-s + (0.950 − 0.311i)5-s + (0.285 + 0.599i)6-s + (−0.999 + 0.00544i)7-s + (−0.732 − 0.732i)8-s + (−0.156 + 0.987i)9-s + (0.663 − 0.0363i)10-s + (0.189 + 0.109i)11-s + (−0.101 − 0.549i)12-s + (−0.400 + 0.400i)13-s + (−0.642 − 0.168i)14-s + (0.853 + 0.520i)15-s + (−0.0644 − 0.111i)16-s + (−0.460 − 1.71i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.890 - 0.454i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (32, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.890 - 0.454i)$
$L(1)$  $\approx$  $1.42112 + 0.341437i$
$L(\frac12)$  $\approx$  $1.42112 + 0.341437i$
$L(\frac{3}{2})$   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 \( 1 + (-1.12 - 1.31i)T \)
5 \( 1 + (-2.12 + 0.695i)T \)
7 \( 1 + (2.64 - 0.0144i)T \)
good2 \( 1 + (-0.907 - 0.243i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-0.630 - 0.363i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.44 - 1.44i)T - 13iT^{2} \)
17 \( 1 + (1.90 + 7.09i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.664 + 0.383i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.840 - 3.13i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 4.07T + 29T^{2} \)
31 \( 1 + (0.209 - 0.363i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.63 - 6.08i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 4.44iT - 41T^{2} \)
43 \( 1 + (5.15 - 5.15i)T - 43iT^{2} \)
47 \( 1 + (-6.79 - 1.82i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-5.26 + 1.41i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.807 + 1.39i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.78 - 8.29i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.90 - 1.84i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.06iT - 71T^{2} \)
73 \( 1 + (4.08 + 15.2i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.80 + 3.35i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.83 - 1.83i)T + 83iT^{2} \)
89 \( 1 + (6.94 + 12.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.62 - 5.62i)T + 97iT^{2} \)
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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

−13.64570571732165821770277944797, −13.45246608434879723439844771266, −12.03287200975879893526215087846, −10.19590425849805418294712634333, −9.517793184463093136823809587813, −8.921419409046556646751323945912, −6.86102560316005186398598892256, −5.48659013711093687890217826623, −4.47843809852646598585471751936, −2.91561138072949283237632359234, 2.50341200074928761404236028459, 3.76174904704881444174441147868, 5.76979049273608164238054647992, 6.72344818375139361284662092171, 8.335011247232703253377835206288, 9.267036483539657269055024943080, 10.40224376480948416405712851920, 12.22313725600981408911708607886, 12.86793691825964548245456133164, 13.53305011332370416329137340443

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