Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.796 - 0.604i$
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 + (1.29 + 1.15i)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 + (0.333 + 2.98i)9-s + (0.690 − 0.00558i)10-s + (−0.660 − 0.381i)11-s + (1.02 + 3.13i)12-s + (−2.27 − 2.27i)13-s + (0.481 − 0.660i)14-s + (1.71 − 3.47i)15-s + (1.71 + 2.97i)16-s + (4.69 − 1.25i)17-s + ⋯
L(s)  = 1  + (−0.0565 + 0.210i)2-s + (0.745 + 0.666i)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 + (0.111 + 0.993i)9-s + (0.218 − 0.00176i)10-s + (−0.199 − 0.114i)11-s + (0.297 + 0.904i)12-s + (−0.629 − 0.629i)13-s + (0.128 − 0.176i)14-s + (0.443 − 0.896i)15-s + (0.429 + 0.744i)16-s + (1.13 − 0.305i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.796 - 0.604i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (53, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.796 - 0.604i)$
$L(1)$  $\approx$  $1.18008 + 0.397362i$
$L(\frac12)$  $\approx$  $1.18008 + 0.397362i$
$L(\frac{3}{2})$   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 \( 1 + (-1.29 - 1.15i)T \)
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

−13.91876752498482332662409718364, −12.75830185690678940225280438133, −12.01312504181843529525331999704, −10.44407065377499929562549714218, −9.603494268852655501788282194420, −8.239011714464485638415362350384, −7.58664751525549216395627323024, −5.84731767878069114127869665785, −4.16017424109475977166389761136, −2.81868405973270212235027282105, 2.28310621613916264379906146969, 3.38879092898241778535373175366, 6.12682987935476739963449822422, 6.87960251611126204181177024292, 7.941454918663588828232473532342, 9.615336622328146906268119569126, 10.30973487366147996947878575794, 11.83990522729379786260310652424, 12.35935393441663691522995162607, 13.85452617165873167409765584376

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