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Properties

Label 1-42e2-1764.239-r0-0-0
Degree $1$
Conductor $1764$
Sign $-0.763 + 0.645i$
Analytic cond. $8.19198$
Root an. cond. $8.19198$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.365 + 0.930i)5-s + (0.955 + 0.294i)11-s + (−0.733 + 0.680i)13-s + (0.900 + 0.433i)17-s − 19-s + (0.826 + 0.563i)23-s + (−0.733 − 0.680i)25-s + (−0.826 + 0.563i)29-s + (0.5 + 0.866i)31-s + (−0.900 − 0.433i)37-s + (−0.365 + 0.930i)41-s + (−0.365 − 0.930i)43-s + (0.955 + 0.294i)47-s + (0.900 − 0.433i)53-s + (−0.623 + 0.781i)55-s + ⋯
L(s)  = 1  + (−0.365 + 0.930i)5-s + (0.955 + 0.294i)11-s + (−0.733 + 0.680i)13-s + (0.900 + 0.433i)17-s − 19-s + (0.826 + 0.563i)23-s + (−0.733 − 0.680i)25-s + (−0.826 + 0.563i)29-s + (0.5 + 0.866i)31-s + (−0.900 − 0.433i)37-s + (−0.365 + 0.930i)41-s + (−0.365 − 0.930i)43-s + (0.955 + 0.294i)47-s + (0.900 − 0.433i)53-s + (−0.623 + 0.781i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.763 + 0.645i$
Analytic conductor: \(8.19198\)
Root analytic conductor: \(8.19198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (0:\ ),\ -0.763 + 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3654890731 + 0.9986356137i\)
\(L(\frac12)\) \(\approx\) \(0.3654890731 + 0.9986356137i\)
\(L(1)\) \(\approx\) \(0.8794914899 + 0.3401873238i\)
\(L(1)\) \(\approx\) \(0.8794914899 + 0.3401873238i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.365 + 0.930i)T \)
11 \( 1 + (0.955 + 0.294i)T \)
13 \( 1 + (-0.733 + 0.680i)T \)
17 \( 1 + (0.900 + 0.433i)T \)
19 \( 1 - T \)
23 \( 1 + (0.826 + 0.563i)T \)
29 \( 1 + (-0.826 + 0.563i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.900 - 0.433i)T \)
41 \( 1 + (-0.365 + 0.930i)T \)
43 \( 1 + (-0.365 - 0.930i)T \)
47 \( 1 + (0.955 + 0.294i)T \)
53 \( 1 + (0.900 - 0.433i)T \)
59 \( 1 + (-0.988 - 0.149i)T \)
61 \( 1 + (0.826 - 0.563i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.733 - 0.680i)T \)
89 \( 1 + (0.222 - 0.974i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.94175464012187577700576457391, −19.244669581646824968694644461869, −18.72380362750474041973587599036, −17.48243035244321197026504298726, −16.84985093351420479553053672, −16.60307190414631038403079070811, −15.33441235961267357181745258435, −14.98413717742679168382104206118, −13.95965267840446832522471569240, −13.18053966733840541181403147931, −12.34861059637070299446171207612, −11.934069796492281986960354812985, −11.002967523130085214154984710918, −10.051616927907668318753141427605, −9.26639100538110257023697631868, −8.57957446085539515240725248264, −7.81545202319689964668499492564, −7.00300350535148158073378751776, −5.956761988371653332776982317659, −5.18743267149067636771709235369, −4.36278575985705130065793831715, −3.58851566615242951617321529576, −2.513841462462669209999150508192, −1.34982494827597582017762132961, −0.396844153785082114932606643872, 1.39101329879838118614827897243, 2.30117499015311711395001884462, 3.3722183511790330332027870873, 3.98448455631385941400656132746, 4.97850656708059694212921061915, 6.028257537589299758443059437525, 6.95268867382039839804604789955, 7.23964651297472788126558965373, 8.39309242323906732342348384154, 9.188878999354909882172135355, 10.07457726247230650445951755837, 10.71649670411610940744516320775, 11.63871341706707504291561013945, 12.12576764248684241202382731735, 13.02708853599528099415824305819, 14.149595277583500804537838011885, 14.595716740842911178132437800011, 15.1407090351339184921661512188, 16.074595358550216178312341423395, 17.1066106700872394471859218573, 17.32885642857695122087427917687, 18.54918743890754419410109954126, 19.11073836582562559486549555309, 19.55251063397138074942171713070, 20.444190157401669202113319998995

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