Properties

Label 2-42e2-1.1-c1-0-1
Degree $2$
Conductor $1764$
Sign $1$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.74·5-s + 5.29·11-s − 4.24·13-s − 3.74·17-s + 2.82·19-s − 5.29·23-s + 9·25-s − 5.29·29-s + 8.48·31-s + 4·37-s − 3.74·41-s + 8·43-s + 7.48·47-s − 10.5·53-s − 19.7·55-s + 7.48·59-s + 9.89·61-s + 15.8·65-s + 12·67-s + 15.8·71-s + 1.41·73-s − 4·79-s + 14.9·83-s + 14·85-s + 3.74·89-s − 10.5·95-s − 9.89·97-s + ⋯
L(s)  = 1  − 1.67·5-s + 1.59·11-s − 1.17·13-s − 0.907·17-s + 0.648·19-s − 1.10·23-s + 1.80·25-s − 0.982·29-s + 1.52·31-s + 0.657·37-s − 0.584·41-s + 1.21·43-s + 1.09·47-s − 1.45·53-s − 2.66·55-s + 0.974·59-s + 1.26·61-s + 1.96·65-s + 1.46·67-s + 1.88·71-s + 0.165·73-s − 0.450·79-s + 1.64·83-s + 1.51·85-s + 0.396·89-s − 1.08·95-s − 1.00·97-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.092726202\)
\(L(\frac12)\) \(\approx\) \(1.092726202\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.74T + 5T^{2} \)
11 \( 1 - 5.29T + 11T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + 3.74T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 + 5.29T + 23T^{2} \)
29 \( 1 + 5.29T + 29T^{2} \)
31 \( 1 - 8.48T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 3.74T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 7.48T + 47T^{2} \)
53 \( 1 + 10.5T + 53T^{2} \)
59 \( 1 - 7.48T + 59T^{2} \)
61 \( 1 - 9.89T + 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 - 15.8T + 71T^{2} \)
73 \( 1 - 1.41T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 - 3.74T + 89T^{2} \)
97 \( 1 + 9.89T + 97T^{2} \)
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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

−9.282773496830415322230506157505, −8.396671947991046328055960854827, −7.71781959762712130559216376966, −7.02639182725517399338975389344, −6.30759465764849707789931966423, −4.97028710262027576641247607431, −4.14794549815201831052728297319, −3.66013022493088008981619970528, −2.34591842039331784623674710933, −0.70840845505015271111271105892, 0.70840845505015271111271105892, 2.34591842039331784623674710933, 3.66013022493088008981619970528, 4.14794549815201831052728297319, 4.97028710262027576641247607431, 6.30759465764849707789931966423, 7.02639182725517399338975389344, 7.71781959762712130559216376966, 8.396671947991046328055960854827, 9.282773496830415322230506157505

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