Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.611 + 0.791i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·2-s + (1 − 1.41i)3-s + 0.999·4-s + (1.73 + 1.41i)5-s + (−1.73 + 2.44i)6-s + (−1 − 2.44i)7-s + 1.73·8-s + (−1.00 − 2.82i)9-s + (−2.99 − 2.44i)10-s − 2.82i·11-s + (0.999 − 1.41i)12-s + 4·13-s + (1.73 + 4.24i)14-s + (3.73 − 1.03i)15-s − 5·16-s + 2.82i·17-s + ⋯
L(s)  = 1  − 1.22·2-s + (0.577 − 0.816i)3-s + 0.499·4-s + (0.774 + 0.632i)5-s + (−0.707 + 0.999i)6-s + (−0.377 − 0.925i)7-s + 0.612·8-s + (−0.333 − 0.942i)9-s + (−0.948 − 0.774i)10-s − 0.852i·11-s + (0.288 − 0.408i)12-s + 1.10·13-s + (0.462 + 1.13i)14-s + (0.963 − 0.267i)15-s − 1.25·16-s + 0.685i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.611 + 0.791i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (104, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.611 + 0.791i)$
$L(1)$  $\approx$  $0.641892 - 0.315073i$
$L(\frac12)$  $\approx$  $0.641892 - 0.315073i$
$L(\frac{3}{2})$   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 \( 1 + (-1 + 1.41i)T \)
5 \( 1 + (-1.73 - 1.41i)T \)
7 \( 1 + (1 + 2.44i)T \)
good2 \( 1 + 1.73T + 2T^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 - 5.65iT - 29T^{2} \)
31 \( 1 - 9.79iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 + 4.89iT - 43T^{2} \)
47 \( 1 - 2.82iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 + 9.79iT - 61T^{2} \)
67 \( 1 - 4.89iT - 67T^{2} \)
71 \( 1 + 2.82iT - 71T^{2} \)
73 \( 1 + 8T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−13.78953602389992299545556030147, −12.83614226853294304656328156174, −11.01275860875083973410540180929, −10.33768172406589589164881798623, −9.133970793684972294741362444975, −8.283893694847508889487503317621, −7.11828797467679815132894401824, −6.19409117106424406375678282124, −3.43890404297096500369201921575, −1.42578641308288919280803743557, 2.20572488300709862004873935729, 4.45085844427355010240983219569, 5.92651193369505600049189576146, 7.86847010231486027892400878543, 8.848326879158142793499529862373, 9.513682576110652253692021784581, 10.11856356347458545064561785431, 11.50249118646801300083746602450, 13.05572997661255576777711014700, 13.87689206863932239502110591337

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