Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.873 + 0.487i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0799 − 0.298i)2-s + (0.540 − 1.64i)3-s + (1.64 + 0.952i)4-s + (0.596 + 2.15i)5-s + (−0.447 − 0.292i)6-s + (−2.46 − 0.951i)7-s + (0.852 − 0.852i)8-s + (−2.41 − 1.77i)9-s + (0.690 − 0.00558i)10-s + (0.660 + 0.381i)11-s + (2.45 − 2.19i)12-s + (−2.27 − 2.27i)13-s + (−0.481 + 0.660i)14-s + (3.86 + 0.184i)15-s + (1.71 + 2.97i)16-s + (−4.69 + 1.25i)17-s + ⋯
L(s)  = 1  + (0.0565 − 0.210i)2-s + (0.312 − 0.950i)3-s + (0.824 + 0.476i)4-s + (0.266 + 0.963i)5-s + (−0.182 − 0.119i)6-s + (−0.933 − 0.359i)7-s + (0.301 − 0.301i)8-s + (−0.805 − 0.593i)9-s + (0.218 − 0.00176i)10-s + (0.199 + 0.114i)11-s + (0.709 − 0.634i)12-s + (−0.629 − 0.629i)13-s + (−0.128 + 0.176i)14-s + (0.998 + 0.0476i)15-s + (0.429 + 0.744i)16-s + (−1.13 + 0.305i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.873 + 0.487i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (53, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.873 + 0.487i)$
$L(1)$  $\approx$  $1.20091 - 0.312429i$
$L(\frac12)$  $\approx$  $1.20091 - 0.312429i$
$L(\frac{3}{2})$   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 \( 1 + (-0.540 + 1.64i)T \)
5 \( 1 + (-0.596 - 2.15i)T \)
7 \( 1 + (2.46 + 0.951i)T \)
good2 \( 1 + (-0.0799 + 0.298i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-0.660 - 0.381i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.27 + 2.27i)T + 13iT^{2} \)
17 \( 1 + (4.69 - 1.25i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.41 - 0.818i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.39 - 1.98i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 4.94T + 29T^{2} \)
31 \( 1 + (-2.96 + 5.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.41 - 0.915i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 4.35iT - 41T^{2} \)
43 \( 1 + (-2.69 - 2.69i)T + 43iT^{2} \)
47 \( 1 + (-1.10 + 4.14i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.79 + 6.71i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.84 + 6.65i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.19 + 3.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.0126 - 0.0471i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 12.4iT - 71T^{2} \)
73 \( 1 + (1.34 - 0.359i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.66 + 2.11i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.05 - 5.05i)T - 83iT^{2} \)
89 \( 1 + (0.453 + 0.785i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.73 + 3.73i)T - 97iT^{2} \)
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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

−13.25990574135713172067486611127, −12.91926529430470319018353159975, −11.60684736859995403691338299752, −10.74004565413189057121571730964, −9.456995753487573712577871588462, −7.80174373170695742902958436374, −6.92593539995165889238527396882, −6.23670208026834130883249928345, −3.42333967823018597516588761266, −2.36346477388012276580321747064, 2.56624740763306355575309366941, 4.54165344404402797406189813979, 5.70454355536339888806825482306, 6.98524833683424819563357942755, 8.823837211170176472227288131686, 9.415391442225670646725080961264, 10.57182402649436770339405091288, 11.65323785234021771127177527259, 12.87634642352758084308708352960, 14.02182893278758026657414443178

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