Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.662 - 0.749i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.193i·2-s + i·3-s + 1.96·4-s + (−1.48 + 1.67i)5-s − 0.193·6-s i·7-s + 0.768i·8-s − 9-s + (−0.324 − 0.287i)10-s + 2·11-s + 1.96i·12-s − 1.35i·13-s + 0.193·14-s + (−1.67 − 1.48i)15-s + 3.77·16-s − 3.35i·17-s + ⋯
L(s)  = 1  + 0.137i·2-s + 0.577i·3-s + 0.981·4-s + (−0.662 + 0.749i)5-s − 0.0791·6-s − 0.377i·7-s + 0.271i·8-s − 0.333·9-s + (−0.102 − 0.0908i)10-s + 0.603·11-s + 0.566i·12-s − 0.374i·13-s + 0.0518·14-s + (−0.432 − 0.382i)15-s + 0.943·16-s − 0.812i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.662 - 0.749i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (64, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.662 - 0.749i)$
$L(1)$  $\approx$  $1.01753 + 0.458536i$
$L(\frac12)$  $\approx$  $1.01753 + 0.458536i$
$L(\frac{3}{2})$   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 \( 1 - iT \)
5 \( 1 + (1.48 - 1.67i)T \)
7 \( 1 + iT \)
good2 \( 1 - 0.193iT - 2T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 1.35iT - 13T^{2} \)
17 \( 1 + 3.35iT - 17T^{2} \)
19 \( 1 + 5.35T + 19T^{2} \)
23 \( 1 + 4.96iT - 23T^{2} \)
29 \( 1 + 7.92T + 29T^{2} \)
31 \( 1 - 4.57T + 31T^{2} \)
37 \( 1 + 0.775iT - 37T^{2} \)
41 \( 1 - 3.73T + 41T^{2} \)
43 \( 1 - 12.6iT - 43T^{2} \)
47 \( 1 - 9.92iT - 47T^{2} \)
53 \( 1 + 8.57iT - 53T^{2} \)
59 \( 1 - 8.62T + 59T^{2} \)
61 \( 1 + 8.70T + 61T^{2} \)
67 \( 1 - 9.92iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 9.35iT - 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 + 3.22iT - 83T^{2} \)
89 \( 1 + 1.03T + 89T^{2} \)
97 \( 1 + 18.4iT - 97T^{2} \)
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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

−14.42500593966397533566738443130, −12.70914338834609206486145666453, −11.45762514181586991195022458648, −10.94048600234398559991926340665, −9.886200586564330919785774353055, −8.269554578974505741263299090642, −7.14629385346476907324518962572, −6.15715531072572172642212975156, −4.25559555092394245803359709341, −2.81996533810922616835865143105, 1.82985493990099968121407889957, 3.83329502351719482017700569137, 5.74893100189629764222583720633, 6.93928565868959444837122682974, 8.051970160515102683253813684499, 9.128776383263763143259776274522, 10.76797034640784383297070855636, 11.78205617523218564929812759355, 12.33513129041615351075814787645, 13.36413621830796445891106378661

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