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Properties

Label 2-42e2-63.58-c1-0-31
Degree $2$
Conductor $1764$
Sign $-0.804 + 0.594i$
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  + (−0.619 − 1.61i)3-s + (0.119 + 0.207i)5-s + (−2.23 + 2.00i)9-s + (2.56 − 4.43i)11-s + (−2.44 + 4.23i)13-s + (0.260 − 0.321i)15-s + (1.85 + 3.20i)17-s + (1.83 − 3.16i)19-s + (−3.71 − 6.42i)23-s + (2.47 − 4.28i)25-s + (4.62 + 2.36i)27-s + (−1.73 − 3.00i)29-s + 0.717·31-s + (−8.76 − 1.39i)33-s + (−2.30 + 3.98i)37-s + ⋯
L(s)  = 1  + (−0.357 − 0.933i)3-s + (0.0534 + 0.0926i)5-s + (−0.744 + 0.668i)9-s + (0.772 − 1.33i)11-s + (−0.677 + 1.17i)13-s + (0.0673 − 0.0830i)15-s + (0.449 + 0.777i)17-s + (0.419 − 0.727i)19-s + (−0.773 − 1.34i)23-s + (0.494 − 0.856i)25-s + (0.890 + 0.455i)27-s + (−0.321 − 0.557i)29-s + 0.128·31-s + (−1.52 − 0.242i)33-s + (−0.378 + 0.655i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.804 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.804 + 0.594i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.804 + 0.594i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.028657967\)
\(L(\frac12)\) \(\approx\) \(1.028657967\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.619 + 1.61i)T \)
7 \( 1 \)
good5 \( 1 + (-0.119 - 0.207i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.56 + 4.43i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.44 - 4.23i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.85 - 3.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.83 + 3.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.71 + 6.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.73 + 3.00i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.717T + 31T^{2} \)
37 \( 1 + (2.30 - 3.98i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.80 + 4.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.24 + 10.8i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.33T + 47T^{2} \)
53 \( 1 + (-0.471 - 0.816i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 7.57T + 59T^{2} \)
61 \( 1 + 5.50T + 61T^{2} \)
67 \( 1 + 0.660T + 67T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 + (1.83 + 3.16i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 6.22T + 79T^{2} \)
83 \( 1 + (-4.85 - 8.40i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.74 - 6.48i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.57 + 14.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.687651094560097880492449755030, −8.334069949622197483882429241554, −7.23356867925093510058678831783, −6.51949944212428890912153312619, −6.04766957499368974476126487876, −4.98737040995492096261140156225, −3.94236573040263917664202067423, −2.74660023397776883750829563384, −1.72329543742589872329626024080, −0.41373475076277126902076439862, 1.43307704917891739661155804569, 2.99021772153115443763074705324, 3.76802053733497101428400685646, 4.86629990404744904613372812694, 5.31467585307117004967488707231, 6.27537864819452361073606922929, 7.34928424805344862720409889756, 7.928907567312920907567413777061, 9.210621454571124714833636849004, 9.682175578773079478404657559608

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