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Properties

Label 2-42e2-12.11-c1-0-59
Degree $2$
Conductor $1764$
Sign $0.832 + 0.554i$
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  + (1.39 − 0.258i)2-s + (1.86 − 0.719i)4-s − 0.732i·5-s + (2.40 − 1.48i)8-s + (−0.189 − 1.01i)10-s + 2.03·11-s + 1.41·13-s + (2.96 − 2.68i)16-s + 4.19i·17-s + 5.56i·19-s + (−0.526 − 1.36i)20-s + (2.83 − 0.526i)22-s + 2.03·23-s + 4.46·25-s + (1.96 − 0.366i)26-s + ⋯
L(s)  = 1  + (0.983 − 0.183i)2-s + (0.933 − 0.359i)4-s − 0.327i·5-s + (0.851 − 0.524i)8-s + (−0.0599 − 0.321i)10-s + 0.613·11-s + 0.392·13-s + (0.741 − 0.671i)16-s + 1.01i·17-s + 1.27i·19-s + (−0.117 − 0.305i)20-s + (0.603 − 0.112i)22-s + 0.424·23-s + 0.892·25-s + (0.385 − 0.0717i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.832 + 0.554i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.640708343\)
\(L(\frac12)\) \(\approx\) \(3.640708343\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.258i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 0.732iT - 5T^{2} \)
11 \( 1 - 2.03T + 11T^{2} \)
13 \( 1 - 1.41T + 13T^{2} \)
17 \( 1 - 4.19iT - 17T^{2} \)
19 \( 1 - 5.56iT - 19T^{2} \)
23 \( 1 - 2.03T + 23T^{2} \)
29 \( 1 + 5.27iT - 29T^{2} \)
31 \( 1 + 9.63iT - 31T^{2} \)
37 \( 1 - 3.46T + 37T^{2} \)
41 \( 1 + 8.73iT - 41T^{2} \)
43 \( 1 - 7.86iT - 43T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 + 9.14iT - 53T^{2} \)
59 \( 1 - 4.98T + 59T^{2} \)
61 \( 1 + 9.41T + 61T^{2} \)
67 \( 1 - 2.87iT - 67T^{2} \)
71 \( 1 + 9.08T + 71T^{2} \)
73 \( 1 - 1.69T + 73T^{2} \)
79 \( 1 - 4.98iT - 79T^{2} \)
83 \( 1 - 13.6T + 83T^{2} \)
89 \( 1 - 8.73iT - 89T^{2} \)
97 \( 1 - 15.0T + 97T^{2} \)
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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

−9.356417297372977521921894527101, −8.274361509937174008216412440730, −7.63231693517923900139550238637, −6.42334561117508308231669305231, −6.06964220152301209713386073908, −5.07167290063203298502835325608, −4.11119716511244129021829268216, −3.55476535515036239550865521777, −2.24376492406876774383590876955, −1.20232809650163689749947458703, 1.35631345910849485935626352593, 2.81878070681408800785343091390, 3.33128806279082806575991586887, 4.63770901713505312702996610091, 5.06638798869277365009554611762, 6.26397150136886529758670732772, 6.86759075781276268503615807506, 7.41314798383073331938586789998, 8.606327034324542041540126242982, 9.219889610589219666528580610347

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