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  :  $\chi_{105} (8, \cdot )$
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

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

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