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Properties

Label 2-42e2-28.27-c1-0-59
Degree $2$
Conductor $1764$
Sign $0.964 - 0.262i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.476 + 1.33i)2-s + (−1.54 + 1.26i)4-s − 0.509i·5-s + (−2.42 − 1.45i)8-s + (0.678 − 0.243i)10-s − 4.12i·11-s + 3.97i·13-s + (0.778 − 3.92i)16-s − 4.93i·17-s − 5.05·19-s + (0.647 + 0.788i)20-s + (5.49 − 1.96i)22-s + 2.91i·23-s + 4.74·25-s + (−5.29 + 1.89i)26-s + ⋯
L(s)  = 1  + (0.336 + 0.941i)2-s + (−0.772 + 0.634i)4-s − 0.228i·5-s + (−0.857 − 0.513i)8-s + (0.214 − 0.0768i)10-s − 1.24i·11-s + 1.10i·13-s + (0.194 − 0.980i)16-s − 1.19i·17-s − 1.15·19-s + (0.144 + 0.176i)20-s + (1.17 − 0.419i)22-s + 0.608i·23-s + 0.948·25-s + (−1.03 + 0.371i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.964 - 0.262i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.964 - 0.262i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.629757160\)
\(L(\frac12)\) \(\approx\) \(1.629757160\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.476 - 1.33i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 0.509iT - 5T^{2} \)
11 \( 1 + 4.12iT - 11T^{2} \)
13 \( 1 - 3.97iT - 13T^{2} \)
17 \( 1 + 4.93iT - 17T^{2} \)
19 \( 1 + 5.05T + 19T^{2} \)
23 \( 1 - 2.91iT - 23T^{2} \)
29 \( 1 - 5.82T + 29T^{2} \)
31 \( 1 - 5.71T + 31T^{2} \)
37 \( 1 - 10.0T + 37T^{2} \)
41 \( 1 + 11.7iT - 41T^{2} \)
43 \( 1 + 9.84iT - 43T^{2} \)
47 \( 1 + 7.87T + 47T^{2} \)
53 \( 1 + 1.23T + 53T^{2} \)
59 \( 1 - 4.65T + 59T^{2} \)
61 \( 1 - 0.543iT - 61T^{2} \)
67 \( 1 + 7.02iT - 67T^{2} \)
71 \( 1 - 1.06iT - 71T^{2} \)
73 \( 1 + 0.837iT - 73T^{2} \)
79 \( 1 - 1.25iT - 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 - 4.68iT - 89T^{2} \)
97 \( 1 + 1.80iT - 97T^{2} \)
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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

−8.945340499096159017403646128094, −8.620668241859990974310576631737, −7.70586653441415216570866297665, −6.75642966953741535515186429800, −6.28092337773447573281566364844, −5.26514739870316262729220210208, −4.54961714405367159371859014529, −3.64403577716071691747339573033, −2.56011279356042462776664656845, −0.64464428242761397595903207352, 1.13624553834073537928284793577, 2.38085281326254528862695528430, 3.12040658355300240441602720277, 4.42860896452501087087438586653, 4.72753812149164536758760904099, 6.08614858700899196319593247812, 6.57570352144186636322491907400, 8.032316397084155711607319628524, 8.410463888709847624516011883655, 9.618677686921470910046001764809

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