Properties

Label 1-42e2-1764.1219-r1-0-0
Degree $1$
Conductor $1764$
Sign $0.645 + 0.763i$
Analytic cond. $189.568$
Root an. cond. $189.568$
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

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

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.645 + 0.763i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ 0.645 + 0.763i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5952472604 + 0.2762788089i\)
\(L(\frac12)\) \(\approx\) \(0.5952472604 + 0.2762788089i\)
\(L(1)\) \(\approx\) \(0.8143098040 - 0.1689138769i\)
\(L(1)\) \(\approx\) \(0.8143098040 - 0.1689138769i\)

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

−19.76849066522484713767485976578, −19.3173366864242925768564949814, −18.34819863226096710314231832546, −17.73279125060745941392537778500, −17.32404758423935844587602409017, −16.15024669840385791470696857331, −15.33209617150260654808885955489, −14.90230825874621230599357247450, −14.055654281887100182829686438554, −13.17617690441861884193185568724, −12.682896815839562026577228204677, −11.57473435473731549543774628673, −10.82859336988411133047224523884, −10.14064531400367214270923981413, −9.6530433923476080940539057337, −8.32839775601866136059774120356, −7.81062211451539442462027800803, −6.79044931130267545349431798378, −6.21994478056954514106064996334, −5.23536942696003840669731379451, −4.41488385924076502709482688045, −3.2674895375583797501073208372, −2.50429457086204764234918333969, −1.80180783623647891163787938899, −0.16660900347813844966016008768, 0.6061227215935569157781328319, 2.01520369286237541182200317868, 2.4553564700481567278758060492, 3.8994261930232180166684751070, 4.72533246326027502390997808226, 5.27845183057285947584570456495, 6.32423762921628181537180175058, 7.0519050625170918523203742360, 8.28389245784917803460066544947, 8.588636299750777585455445584091, 9.60032214569249709507897470469, 10.271798697838504678948386765516, 11.11467581671082813452064659203, 12.19014787420126788454110183276, 12.55724956551400495456920741289, 13.58713903534523473192986424515, 13.95978042934317028447345149847, 15.04277931781398449097797300401, 15.998615483595886693671865323018, 16.26829222127061832722307750616, 17.46386017865616300854002162833, 17.614185383593970930014901708572, 18.81007472258106370081921539716, 19.4084539119179694549065097616, 20.25250206384776866888432439010

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