Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.0101 + 0.999i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.582 − 2.17i)2-s + (1.60 + 0.644i)3-s + (−2.64 + 1.52i)4-s + (2.21 + 0.337i)5-s + (0.465 − 3.86i)6-s + (−1.15 − 2.38i)7-s + (1.68 + 1.68i)8-s + (2.16 + 2.07i)9-s + (−0.552 − 4.99i)10-s + (−3.88 + 2.24i)11-s + (−5.24 + 0.750i)12-s + (−1.08 + 1.08i)13-s + (−4.50 + 3.88i)14-s + (3.33 + 1.96i)15-s + (−0.381 + 0.660i)16-s + (2.04 + 0.548i)17-s + ⋯
L(s)  = 1  + (−0.411 − 1.53i)2-s + (0.928 + 0.372i)3-s + (−1.32 + 0.764i)4-s + (0.988 + 0.151i)5-s + (0.189 − 1.57i)6-s + (−0.435 − 0.900i)7-s + (0.595 + 0.595i)8-s + (0.722 + 0.691i)9-s + (−0.174 − 1.58i)10-s + (−1.17 + 0.676i)11-s + (−1.51 + 0.216i)12-s + (−0.300 + 0.300i)13-s + (−1.20 + 1.03i)14-s + (0.861 + 0.508i)15-s + (−0.0952 + 0.165i)16-s + (0.496 + 0.133i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0101 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0101 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.0101 + 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.0101 + 0.999i)$
$L(1)$  $\approx$  $0.779116 - 0.771211i$
$L(\frac12)$  $\approx$  $0.779116 - 0.771211i$
$L(\frac{3}{2})$   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 \( 1 + (-1.60 - 0.644i)T \)
5 \( 1 + (-2.21 - 0.337i)T \)
7 \( 1 + (1.15 + 2.38i)T \)
good2 \( 1 + (0.582 + 2.17i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (3.88 - 2.24i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.08 - 1.08i)T - 13iT^{2} \)
17 \( 1 + (-2.04 - 0.548i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.66 + 2.11i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.13 + 0.840i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 1.69T + 29T^{2} \)
31 \( 1 + (-0.530 - 0.918i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.75 - 1.54i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 5.84iT - 41T^{2} \)
43 \( 1 + (-2.00 + 2.00i)T - 43iT^{2} \)
47 \( 1 + (1.36 + 5.10i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.23 + 8.34i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.35 + 4.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.88 + 6.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.152 - 0.569i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.66iT - 71T^{2} \)
73 \( 1 + (4.22 + 1.13i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.78 - 3.33i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.0 + 11.0i)T + 83iT^{2} \)
89 \( 1 + (1.75 - 3.04i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.60 + 5.60i)T + 97iT^{2} \)
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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

−13.15379137399328579084558878767, −12.77666842443899308348240594996, −10.94313095017620630843059114488, −10.15016204953519607850270152674, −9.733818300614316018014726331490, −8.552805211039367982820626465671, −7.03848786680146135277330862342, −4.71671659878680323709511292350, −3.20740138998166798676052528207, −2.04160873205741505505109639699, 2.68273035298435408457039316310, 5.34964820489348257203403448868, 6.19352223653619115902307503916, 7.47263808704472560034095325748, 8.521487928959670552542380451496, 9.189573307143233919270094558080, 10.26381501969504394658924651200, 12.49676875160136887930819857577, 13.34943715623176579031054172635, 14.24645229814027447799402009826

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