Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.443 - 0.896i$
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 + (−1.75 − 1.97i)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 + (−0.665 − 0.746i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.443 - 0.896i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (97, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.443 - 0.896i)$
$L(1)$  $\approx$  $1.26862 + 0.787305i$
$L(\frac12)$  $\approx$  $1.26862 + 0.787305i$
$L(\frac{3}{2})$   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 \( 1 + (1.75 + 1.97i)T \)
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

−13.96575362278167076398685625496, −13.20461508243351978949758959535, −12.39985170499961092071568700970, −10.74280493177525055304710924216, −9.926152481612053452362094039768, −7.75473144207418836566183568892, −6.94449627271380016053651299234, −6.07544782999447169228396953131, −4.94469544713314655053931170402, −3.14774952490807975471958947124, 2.26675257482203083097242262209, 3.96351246892718436237204515860, 5.25306099255159287900039688562, 6.03014666693700717711792218993, 8.382089878248136318605933992594, 9.769631663070175619063329750045, 10.50253493954416741366167364304, 11.80065339443552993838825825667, 12.70204894180904857698640093866, 13.02825583424224659807547311040

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