Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.634 + 0.772i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.347 − 0.347i)2-s + (0.176 − 1.72i)3-s + 1.75i·4-s + (1.16 − 1.90i)5-s + (−0.536 − 0.659i)6-s + (−0.707 − 0.707i)7-s + (1.30 + 1.30i)8-s + (−2.93 − 0.607i)9-s + (−0.256 − 1.06i)10-s + 2.67i·11-s + (3.03 + 0.310i)12-s + (2.14 − 2.14i)13-s − 0.490·14-s + (−3.07 − 2.34i)15-s − 2.61·16-s + (−3.26 + 3.26i)17-s + ⋯
L(s)  = 1  + (0.245 − 0.245i)2-s + (0.101 − 0.994i)3-s + 0.879i·4-s + (0.522 − 0.852i)5-s + (−0.219 − 0.269i)6-s + (−0.267 − 0.267i)7-s + (0.461 + 0.461i)8-s + (−0.979 − 0.202i)9-s + (−0.0810 − 0.337i)10-s + 0.805i·11-s + (0.874 + 0.0895i)12-s + (0.596 − 0.596i)13-s − 0.131·14-s + (−0.795 − 0.606i)15-s − 0.653·16-s + (−0.792 + 0.792i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.634 + 0.772i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (92, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.634 + 0.772i)$
$L(1)$  $\approx$  $1.09251 - 0.516598i$
$L(\frac12)$  $\approx$  $1.09251 - 0.516598i$
$L(\frac{3}{2})$   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 \( 1 + (-0.176 + 1.72i)T \)
5 \( 1 + (-1.16 + 1.90i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (-0.347 + 0.347i)T - 2iT^{2} \)
11 \( 1 - 2.67iT - 11T^{2} \)
13 \( 1 + (-2.14 + 2.14i)T - 13iT^{2} \)
17 \( 1 + (3.26 - 3.26i)T - 17iT^{2} \)
19 \( 1 - 5.24iT - 19T^{2} \)
23 \( 1 + (-2.54 - 2.54i)T + 23iT^{2} \)
29 \( 1 - 2.86T + 29T^{2} \)
31 \( 1 + 5.28T + 31T^{2} \)
37 \( 1 + (2.14 + 2.14i)T + 37iT^{2} \)
41 \( 1 + 11.5iT - 41T^{2} \)
43 \( 1 + (-0.759 + 0.759i)T - 43iT^{2} \)
47 \( 1 + (-7.66 + 7.66i)T - 47iT^{2} \)
53 \( 1 + (4.43 + 4.43i)T + 53iT^{2} \)
59 \( 1 + 0.159T + 59T^{2} \)
61 \( 1 - 4.72T + 61T^{2} \)
67 \( 1 + (5.41 + 5.41i)T + 67iT^{2} \)
71 \( 1 - 13.5iT - 71T^{2} \)
73 \( 1 + (-4.16 + 4.16i)T - 73iT^{2} \)
79 \( 1 - 3.89iT - 79T^{2} \)
83 \( 1 + (4.03 + 4.03i)T + 83iT^{2} \)
89 \( 1 + 3.95T + 89T^{2} \)
97 \( 1 + (1.86 + 1.86i)T + 97iT^{2} \)
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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

−13.23806933126271267357114288223, −12.73195638420862697815409659419, −12.02384622685424869074877174625, −10.61164957956366329633708048896, −8.997805312189156157665155436044, −8.134922109747898602213332364036, −7.00473770034724808110455540459, −5.56335218045376400600811067595, −3.81378935783300803668452714187, −1.95593745688978063229320285775, 2.84756299872267457114284816361, 4.62151244137868787217991599147, 5.88584433572898520220544980304, 6.78416364138023333545945051806, 8.919165530147509188709391781762, 9.616297132950294685728721646770, 10.87670398118492051583745963281, 11.22466996838569723204127017975, 13.43744941006863907659783704560, 14.02926795452803656667032623387

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