Properties

Label 2-42e2-1.1-c3-0-45
Degree $2$
Conductor $1764$
Sign $-1$
Analytic cond. $104.079$
Root an. cond. $10.2019$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 14·5-s − 4·11-s − 54·13-s − 14·17-s − 92·19-s + 152·23-s + 71·25-s + 106·29-s + 144·31-s + 158·37-s − 390·41-s − 508·43-s − 528·47-s − 606·53-s − 56·55-s − 364·59-s − 678·61-s − 756·65-s + 844·67-s + 8·71-s + 422·73-s + 384·79-s − 548·83-s − 196·85-s + 1.19e3·89-s − 1.28e3·95-s + 1.50e3·97-s + ⋯
L(s)  = 1  + 1.25·5-s − 0.109·11-s − 1.15·13-s − 0.199·17-s − 1.11·19-s + 1.37·23-s + 0.567·25-s + 0.678·29-s + 0.834·31-s + 0.702·37-s − 1.48·41-s − 1.80·43-s − 1.63·47-s − 1.57·53-s − 0.137·55-s − 0.803·59-s − 1.42·61-s − 1.44·65-s + 1.53·67-s + 0.0133·71-s + 0.676·73-s + 0.546·79-s − 0.724·83-s − 0.250·85-s + 1.42·89-s − 1.39·95-s + 1.57·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(104.079\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1764} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 14 T + p^{3} T^{2} \)
11 \( 1 + 4 T + p^{3} T^{2} \)
13 \( 1 + 54 T + p^{3} T^{2} \)
17 \( 1 + 14 T + p^{3} T^{2} \)
19 \( 1 + 92 T + p^{3} T^{2} \)
23 \( 1 - 152 T + p^{3} T^{2} \)
29 \( 1 - 106 T + p^{3} T^{2} \)
31 \( 1 - 144 T + p^{3} T^{2} \)
37 \( 1 - 158 T + p^{3} T^{2} \)
41 \( 1 + 390 T + p^{3} T^{2} \)
43 \( 1 + 508 T + p^{3} T^{2} \)
47 \( 1 + 528 T + p^{3} T^{2} \)
53 \( 1 + 606 T + p^{3} T^{2} \)
59 \( 1 + 364 T + p^{3} T^{2} \)
61 \( 1 + 678 T + p^{3} T^{2} \)
67 \( 1 - 844 T + p^{3} T^{2} \)
71 \( 1 - 8 T + p^{3} T^{2} \)
73 \( 1 - 422 T + p^{3} T^{2} \)
79 \( 1 - 384 T + p^{3} T^{2} \)
83 \( 1 + 548 T + p^{3} T^{2} \)
89 \( 1 - 1194 T + p^{3} T^{2} \)
97 \( 1 - 1502 T + p^{3} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.614467037088178232126517842165, −7.81935535784415108227138387633, −6.59279040186920272429682738196, −6.41038534571295420806227146525, −5.04589330251369409521972065992, −4.79236387925037506460691787587, −3.22268254652689818743861051328, −2.35862291006816767023424067526, −1.48069920034264539280955165435, 0, 1.48069920034264539280955165435, 2.35862291006816767023424067526, 3.22268254652689818743861051328, 4.79236387925037506460691787587, 5.04589330251369409521972065992, 6.41038534571295420806227146525, 6.59279040186920272429682738196, 7.81935535784415108227138387633, 8.614467037088178232126517842165

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