Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.979 - 0.201i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.340 + 1.26i)2-s + (−0.664 − 1.59i)3-s + (0.236 + 0.136i)4-s + (1.25 − 1.85i)5-s + (2.25 − 0.299i)6-s + (2.32 − 1.25i)7-s + (−2.11 + 2.11i)8-s + (−2.11 + 2.12i)9-s + (1.92 + 2.21i)10-s + (3.38 + 1.95i)11-s + (0.0611 − 0.468i)12-s + (−1.56 − 1.56i)13-s + (0.807 + 3.38i)14-s + (−3.79 − 0.767i)15-s + (−1.69 − 2.92i)16-s + (−2.58 + 0.693i)17-s + ⋯
L(s)  = 1  + (−0.240 + 0.897i)2-s + (−0.383 − 0.923i)3-s + (0.118 + 0.0681i)4-s + (0.559 − 0.829i)5-s + (0.921 − 0.122i)6-s + (0.879 − 0.476i)7-s + (−0.746 + 0.746i)8-s + (−0.705 + 0.708i)9-s + (0.609 + 0.701i)10-s + (1.01 + 0.588i)11-s + (0.0176 − 0.135i)12-s + (−0.434 − 0.434i)13-s + (0.215 + 0.903i)14-s + (−0.980 − 0.198i)15-s + (−0.422 − 0.731i)16-s + (−0.627 + 0.168i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.979 - 0.201i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (53, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.979 - 0.201i)$
$L(1)$  $\approx$  $0.977819 + 0.0997019i$
$L(\frac12)$  $\approx$  $0.977819 + 0.0997019i$
$L(\frac{3}{2})$   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 \( 1 + (0.664 + 1.59i)T \)
5 \( 1 + (-1.25 + 1.85i)T \)
7 \( 1 + (-2.32 + 1.25i)T \)
good2 \( 1 + (0.340 - 1.26i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-3.38 - 1.95i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.56 + 1.56i)T + 13iT^{2} \)
17 \( 1 + (2.58 - 0.693i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.61 - 0.930i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.38 + 0.638i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 0.513T + 29T^{2} \)
31 \( 1 + (4.29 - 7.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.60 + 1.77i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.308iT - 41T^{2} \)
43 \( 1 + (-7.60 - 7.60i)T + 43iT^{2} \)
47 \( 1 + (1.36 - 5.10i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.498 - 1.85i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.259 + 0.448i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.55 + 4.42i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.34 - 8.74i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 15.3iT - 71T^{2} \)
73 \( 1 + (-2.79 + 0.749i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.37 + 2.52i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.16 + 9.16i)T - 83iT^{2} \)
89 \( 1 + (5.67 + 9.82i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.81 - 6.81i)T - 97iT^{2} \)
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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

−14.00467130269695573303423589526, −12.65220094476052329926805090928, −11.97811278527315043932502869841, −10.76990987310771620923462650517, −9.035451114484931641869710248700, −8.083905201397075981553730335517, −7.10607630303252574186231754349, −6.06944224308059868520903317207, −4.84054053269097728575536111184, −1.82756279041259597197516985663, 2.24905959127290116076869027754, 3.87371796419549271614069142140, 5.64653931949863025770902599871, 6.72755079902891027125056980273, 8.885571103154160664878071026267, 9.672886849462221488776597390094, 10.78576093012541445434214948499, 11.34664175645738743097367701244, 12.09971993693210005276973881984, 13.93215871242626676792513798694

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