Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.718 + 0.695i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.24 − 1.24i)2-s + (1.66 + 0.474i)3-s + 1.09i·4-s + (1.67 + 1.48i)5-s + (−1.48 − 2.66i)6-s + (0.707 − 0.707i)7-s + (−1.12 + 1.12i)8-s + (2.54 + 1.58i)9-s + (−0.241 − 3.92i)10-s − 1.55i·11-s + (−0.520 + 1.82i)12-s + (−4.50 − 4.50i)13-s − 1.75·14-s + (2.08 + 3.26i)15-s + 4.99·16-s + (−2.13 − 2.13i)17-s + ⋯
L(s)  = 1  + (−0.879 − 0.879i)2-s + (0.961 + 0.274i)3-s + 0.547i·4-s + (0.749 + 0.662i)5-s + (−0.604 − 1.08i)6-s + (0.267 − 0.267i)7-s + (−0.397 + 0.397i)8-s + (0.849 + 0.527i)9-s + (−0.0763 − 1.24i)10-s − 0.468i·11-s + (−0.150 + 0.526i)12-s + (−1.25 − 1.25i)13-s − 0.470·14-s + (0.539 + 0.842i)15-s + 1.24·16-s + (−0.517 − 0.517i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.718 + 0.695i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (8, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.718 + 0.695i)$
$L(1)$  $\approx$  $0.878679 - 0.355916i$
$L(\frac12)$  $\approx$  $0.878679 - 0.355916i$
$L(\frac{3}{2})$   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 \( 1 + (-1.66 - 0.474i)T \)
5 \( 1 + (-1.67 - 1.48i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (1.24 + 1.24i)T + 2iT^{2} \)
11 \( 1 + 1.55iT - 11T^{2} \)
13 \( 1 + (4.50 + 4.50i)T + 13iT^{2} \)
17 \( 1 + (2.13 + 2.13i)T + 17iT^{2} \)
19 \( 1 - 4.20iT - 19T^{2} \)
23 \( 1 + (3.76 - 3.76i)T - 23iT^{2} \)
29 \( 1 - 2.97T + 29T^{2} \)
31 \( 1 + 5.79T + 31T^{2} \)
37 \( 1 + (1.23 - 1.23i)T - 37iT^{2} \)
41 \( 1 + 2.68iT - 41T^{2} \)
43 \( 1 + (2.09 + 2.09i)T + 43iT^{2} \)
47 \( 1 + (0.0358 + 0.0358i)T + 47iT^{2} \)
53 \( 1 + (-4.30 + 4.30i)T - 53iT^{2} \)
59 \( 1 + 4.93T + 59T^{2} \)
61 \( 1 - 3.31T + 61T^{2} \)
67 \( 1 + (-1.71 + 1.71i)T - 67iT^{2} \)
71 \( 1 + 5.73iT - 71T^{2} \)
73 \( 1 + (-7.26 - 7.26i)T + 73iT^{2} \)
79 \( 1 - 3.59iT - 79T^{2} \)
83 \( 1 + (12.2 - 12.2i)T - 83iT^{2} \)
89 \( 1 + 1.35T + 89T^{2} \)
97 \( 1 + (-10.9 + 10.9i)T - 97iT^{2} \)
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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

−13.79866935796095075858552677234, −12.50966536531819622678954284352, −11.09649589834245915128172373294, −10.14189709770643150027563165253, −9.719397709357692197609301053482, −8.460737834443961488174606895098, −7.40105416674908944301650870829, −5.46704660621550530269247430054, −3.26506566590548787064319457337, −2.08071031822733052795428344790, 2.15500145594797824474335887863, 4.56609156462555077430029068197, 6.43528230143014186094896716894, 7.38769146631104065035399429068, 8.597893986321220014526337398381, 9.199087651824277501250314245971, 10.02090516786497657463231119193, 12.12028476409280020782717094089, 12.93858475593773229843968348819, 14.17466399929543391012518571067

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