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  :  $\chi_{105} (2, \cdot )$
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

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

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