Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.312 + 0.949i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.260 − 0.260i)2-s + (−1.52 + 0.826i)3-s − 1.86i·4-s + (0.895 − 2.04i)5-s + (0.611 + 0.180i)6-s + (0.707 − 0.707i)7-s + (−1.00 + 1.00i)8-s + (1.63 − 2.51i)9-s + (−0.766 + 0.300i)10-s − 3.38i·11-s + (1.54 + 2.83i)12-s + (1.59 + 1.59i)13-s − 0.368·14-s + (0.331 + 3.85i)15-s − 3.20·16-s + (−0.140 − 0.140i)17-s + ⋯
L(s)  = 1  + (−0.184 − 0.184i)2-s + (−0.878 + 0.477i)3-s − 0.932i·4-s + (0.400 − 0.916i)5-s + (0.249 + 0.0738i)6-s + (0.267 − 0.267i)7-s + (−0.355 + 0.355i)8-s + (0.544 − 0.838i)9-s + (−0.242 + 0.0949i)10-s − 1.02i·11-s + (0.445 + 0.819i)12-s + (0.442 + 0.442i)13-s − 0.0983·14-s + (0.0856 + 0.996i)15-s − 0.801·16-s + (−0.0341 − 0.0341i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.312 + 0.949i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (8, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.312 + 0.949i)$
$L(1)$  $\approx$  $0.622458 - 0.450597i$
$L(\frac12)$  $\approx$  $0.622458 - 0.450597i$
$L(\frac{3}{2})$   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 \( 1 + (1.52 - 0.826i)T \)
5 \( 1 + (-0.895 + 2.04i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (0.260 + 0.260i)T + 2iT^{2} \)
11 \( 1 + 3.38iT - 11T^{2} \)
13 \( 1 + (-1.59 - 1.59i)T + 13iT^{2} \)
17 \( 1 + (0.140 + 0.140i)T + 17iT^{2} \)
19 \( 1 - 7.34iT - 19T^{2} \)
23 \( 1 + (2.21 - 2.21i)T - 23iT^{2} \)
29 \( 1 - 9.49T + 29T^{2} \)
31 \( 1 - 0.922T + 31T^{2} \)
37 \( 1 + (-5.91 + 5.91i)T - 37iT^{2} \)
41 \( 1 - 1.39iT - 41T^{2} \)
43 \( 1 + (-0.864 - 0.864i)T + 43iT^{2} \)
47 \( 1 + (0.651 + 0.651i)T + 47iT^{2} \)
53 \( 1 + (6.54 - 6.54i)T - 53iT^{2} \)
59 \( 1 - 6.25T + 59T^{2} \)
61 \( 1 - 1.83T + 61T^{2} \)
67 \( 1 + (0.815 - 0.815i)T - 67iT^{2} \)
71 \( 1 - 9.77iT - 71T^{2} \)
73 \( 1 + (4.80 + 4.80i)T + 73iT^{2} \)
79 \( 1 - 3.41iT - 79T^{2} \)
83 \( 1 + (-6.26 + 6.26i)T - 83iT^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + (6.71 - 6.71i)T - 97iT^{2} \)
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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

−13.66522515172413528401573858427, −12.26726503752485071737616983027, −11.32514018465127694558394868569, −10.36297294503235450212508593587, −9.532027676961142401876251197505, −8.376237314321435571474218746225, −6.23995639161911179930212974123, −5.56089350016726514038977021190, −4.26882470716815596049442126676, −1.19412052019984611070403396522, 2.59320498130693397251481160305, 4.68625911411458462827976759810, 6.37975764232297280139446511164, 7.11472540598666378699804254213, 8.261629817136296311855082276327, 9.833284419992469923974818306733, 10.99692260095825073694698949091, 11.88147616964867743209516145092, 12.84892561097150354157455622304, 13.73021396133701210524849745677

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