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Properties

Label 2-42e2-7.6-c2-0-3
Degree $2$
Conductor $1764$
Sign $-0.755 - 0.654i$
Analytic cond. $48.0655$
Root an. cond. $6.93293$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 25.9i·13-s − 8.66i·19-s + 25·25-s − 19.0i·31-s − 47·37-s − 83·43-s + 96.9i·61-s − 109·67-s − 29.4i·73-s + 131·79-s + 193. i·97-s + 202. i·103-s − 143·109-s + ⋯
L(s)  = 1  + 1.99i·13-s − 0.455i·19-s + 25-s − 0.614i·31-s − 1.27·37-s − 1.93·43-s + 1.59i·61-s − 1.62·67-s − 0.403i·73-s + 1.65·79-s + 1.99i·97-s + 1.96i·103-s − 1.31·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.755 - 0.654i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ -0.755 - 0.654i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9302248497\)
\(L(\frac12)\) \(\approx\) \(0.9302248497\)
\(L(2)\) 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 \( 1 - 25T^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 - 25.9iT - 169T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 + 8.66iT - 361T^{2} \)
23 \( 1 + 529T^{2} \)
29 \( 1 + 841T^{2} \)
31 \( 1 + 19.0iT - 961T^{2} \)
37 \( 1 + 47T + 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 83T + 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 - 96.9iT - 3.72e3T^{2} \)
67 \( 1 + 109T + 4.48e3T^{2} \)
71 \( 1 + 5.04e3T^{2} \)
73 \( 1 + 29.4iT - 5.32e3T^{2} \)
79 \( 1 - 131T + 6.24e3T^{2} \)
83 \( 1 - 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 - 193. iT - 9.40e3T^{2} \)
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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

−9.167619524977627984786768467535, −8.893507568642603225504536023398, −7.82371266056352850905535584450, −6.84246275220602777596355889579, −6.49502437640926880484117657050, −5.23686254162897457698565427796, −4.49605238053017678983952142430, −3.60160788491019171088966842766, −2.40210214058272971773926815179, −1.39437209184315243454134998295, 0.24362651316094292010829520575, 1.51759060196703476486870935817, 2.92255987812982272530874233845, 3.53004825071167767830153317139, 4.88478739772536426627467866995, 5.44770955865084735282240718119, 6.40687313990306286913915338816, 7.26591815117370438036045134988, 8.138245616333137025369813177963, 8.618561865501307005566447917880

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