Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.952 + 0.306i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.800 + 0.800i)2-s + (−1.09 − 1.34i)3-s − 0.718i·4-s + (2.10 − 0.754i)5-s + (0.199 − 1.95i)6-s + (−0.707 + 0.707i)7-s + (2.17 − 2.17i)8-s + (−0.606 + 2.93i)9-s + (2.28 + 1.08i)10-s + 5.20i·11-s + (−0.964 + 0.785i)12-s + (−3.24 − 3.24i)13-s − 1.13·14-s + (−3.31 − 2.00i)15-s + 2.04·16-s + (0.844 + 0.844i)17-s + ⋯
L(s)  = 1  + (0.566 + 0.566i)2-s + (−0.631 − 0.775i)3-s − 0.359i·4-s + (0.941 − 0.337i)5-s + (0.0813 − 0.796i)6-s + (−0.267 + 0.267i)7-s + (0.769 − 0.769i)8-s + (−0.202 + 0.979i)9-s + (0.723 + 0.341i)10-s + 1.56i·11-s + (−0.278 + 0.226i)12-s + (−0.900 − 0.900i)13-s − 0.302·14-s + (−0.856 − 0.516i)15-s + 0.511·16-s + (0.204 + 0.204i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.952 + 0.306i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (8, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.952 + 0.306i)$
$L(1)$  $\approx$  $1.20946 - 0.189608i$
$L(\frac12)$  $\approx$  $1.20946 - 0.189608i$
$L(\frac{3}{2})$   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 \( 1 + (1.09 + 1.34i)T \)
5 \( 1 + (-2.10 + 0.754i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (-0.800 - 0.800i)T + 2iT^{2} \)
11 \( 1 - 5.20iT - 11T^{2} \)
13 \( 1 + (3.24 + 3.24i)T + 13iT^{2} \)
17 \( 1 + (-0.844 - 0.844i)T + 17iT^{2} \)
19 \( 1 - 1.32iT - 19T^{2} \)
23 \( 1 + (5.62 - 5.62i)T - 23iT^{2} \)
29 \( 1 - 4.38T + 29T^{2} \)
31 \( 1 + 1.70T + 31T^{2} \)
37 \( 1 + (1.71 - 1.71i)T - 37iT^{2} \)
41 \( 1 - 1.82iT - 41T^{2} \)
43 \( 1 + (0.281 + 0.281i)T + 43iT^{2} \)
47 \( 1 + (3.39 + 3.39i)T + 47iT^{2} \)
53 \( 1 + (-3.51 + 3.51i)T - 53iT^{2} \)
59 \( 1 - 1.81T + 59T^{2} \)
61 \( 1 + 2.47T + 61T^{2} \)
67 \( 1 + (-7.92 + 7.92i)T - 67iT^{2} \)
71 \( 1 + 9.06iT - 71T^{2} \)
73 \( 1 + (1.33 + 1.33i)T + 73iT^{2} \)
79 \( 1 + 11.5iT - 79T^{2} \)
83 \( 1 + (5.46 - 5.46i)T - 83iT^{2} \)
89 \( 1 + 9.43T + 89T^{2} \)
97 \( 1 + (3.06 - 3.06i)T - 97iT^{2} \)
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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

−13.64694242910135106182736411076, −12.78074681911096866117436862216, −12.12393888649730160483569988036, −10.29175454439059112760219584415, −9.756157406457628016915272588255, −7.75699758381480193502843083159, −6.69171445998239467331410681333, −5.66093781661780693293120598090, −4.87718749030754123064819634956, −1.88209981289986690488224305437, 2.81380028136633989558056339741, 4.20978539455641778155056428383, 5.53571340099068463547345076261, 6.74497994914376859413233139139, 8.633888192996722617610450202862, 9.886907270414680614812934020707, 10.80475995513307109176466318938, 11.66010697887623285967427188563, 12.66752643302369429647103741802, 13.93692566294017443107492849548

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