Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.337 + 0.941i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.48 − 1.48i)2-s + (0.707 − 0.707i)3-s − 2.43i·4-s + (−1.28 + 1.82i)5-s − 2.10i·6-s + (−1.97 + 1.75i)7-s + (−0.640 − 0.640i)8-s − 1.00i·9-s + (0.798 + 4.63i)10-s − 2.67·11-s + (−1.71 − 1.71i)12-s + (1.22 − 1.22i)13-s + (−0.320 + 5.55i)14-s + (0.379 + 2.20i)15-s + 2.95·16-s + (−4.74 − 4.74i)17-s + ⋯
L(s)  = 1  + (1.05 − 1.05i)2-s + (0.408 − 0.408i)3-s − 1.21i·4-s + (−0.576 + 0.816i)5-s − 0.859i·6-s + (−0.746 + 0.665i)7-s + (−0.226 − 0.226i)8-s − 0.333i·9-s + (0.252 + 1.46i)10-s − 0.805·11-s + (−0.496 − 0.496i)12-s + (0.340 − 0.340i)13-s + (−0.0857 + 1.48i)14-s + (0.0979 + 0.568i)15-s + 0.738·16-s + (−1.15 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.337 + 0.941i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.337 + 0.941i)$
$L(1)$  $\approx$  $1.32362 - 0.931179i$
$L(\frac12)$  $\approx$  $1.32362 - 0.931179i$
$L(\frac{3}{2})$   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 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (1.28 - 1.82i)T \)
7 \( 1 + (1.97 - 1.75i)T \)
good2 \( 1 + (-1.48 + 1.48i)T - 2iT^{2} \)
11 \( 1 + 2.67T + 11T^{2} \)
13 \( 1 + (-1.22 + 1.22i)T - 13iT^{2} \)
17 \( 1 + (4.74 + 4.74i)T + 17iT^{2} \)
19 \( 1 - 6.01T + 19T^{2} \)
23 \( 1 + (0.175 + 0.175i)T + 23iT^{2} \)
29 \( 1 - 0.304iT - 29T^{2} \)
31 \( 1 - 7.25iT - 31T^{2} \)
37 \( 1 + (0.735 - 0.735i)T - 37iT^{2} \)
41 \( 1 - 7.05iT - 41T^{2} \)
43 \( 1 + (-0.304 - 0.304i)T + 43iT^{2} \)
47 \( 1 + (-0.556 - 0.556i)T + 47iT^{2} \)
53 \( 1 + (4.99 + 4.99i)T + 53iT^{2} \)
59 \( 1 + 7.98T + 59T^{2} \)
61 \( 1 + 5.53iT - 61T^{2} \)
67 \( 1 + (3.43 - 3.43i)T - 67iT^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 + (-10.0 + 10.0i)T - 73iT^{2} \)
79 \( 1 - 11.2iT - 79T^{2} \)
83 \( 1 + (4.88 - 4.88i)T - 83iT^{2} \)
89 \( 1 - 6.91T + 89T^{2} \)
97 \( 1 + (8.84 + 8.84i)T + 97iT^{2} \)
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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

−13.46154328281117603358787759231, −12.53961046913951771898762856177, −11.67716262393124706241514880682, −10.80102208899077572231836963130, −9.556748656941572944039734333971, −7.976439580007997460426167251091, −6.65310141686868458159745451163, −5.10795355781075950167428563655, −3.35043746756677708211806738135, −2.63722623679570842402095680740, 3.63815970734262418114012920884, 4.52566424398998115943820254710, 5.82382444740674872213383402491, 7.22288484358396964772399278840, 8.171845937891230575547335880387, 9.495322756084069321983760233535, 10.85518557524411744895454370186, 12.43248060287353479718328709874, 13.30932569451950542128459964125, 13.83074503625887424806465336586

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