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Properties

Label 2-42e2-63.41-c1-0-32
Degree $2$
Conductor $1764$
Sign $-0.482 + 0.876i$
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  + (1.20 − 1.24i)3-s + (−1.37 + 2.37i)5-s + (−0.108 − 2.99i)9-s + (0.362 − 0.209i)11-s + (−1.32 − 0.765i)13-s + (1.31 + 4.56i)15-s − 3.90·17-s − 5.91i·19-s + (−7.72 − 4.46i)23-s + (−1.26 − 2.18i)25-s + (−3.86 − 3.47i)27-s + (6.00 − 3.46i)29-s + (3.05 + 1.76i)31-s + (0.174 − 0.703i)33-s + 9.09·37-s + ⋯
L(s)  = 1  + (0.694 − 0.719i)3-s + (−0.613 + 1.06i)5-s + (−0.0360 − 0.999i)9-s + (0.109 − 0.0630i)11-s + (−0.367 − 0.212i)13-s + (0.338 + 1.17i)15-s − 0.947·17-s − 1.35i·19-s + (−1.61 − 0.930i)23-s + (−0.252 − 0.437i)25-s + (−0.744 − 0.667i)27-s + (1.11 − 0.643i)29-s + (0.548 + 0.316i)31-s + (0.0304 − 0.122i)33-s + 1.49·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.482 + 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.482 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.482 + 0.876i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.482 + 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.209341053\)
\(L(\frac12)\) \(\approx\) \(1.209341053\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.20 + 1.24i)T \)
7 \( 1 \)
good5 \( 1 + (1.37 - 2.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.362 + 0.209i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.32 + 0.765i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.90T + 17T^{2} \)
19 \( 1 + 5.91iT - 19T^{2} \)
23 \( 1 + (7.72 + 4.46i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.00 + 3.46i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.05 - 1.76i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.09T + 37T^{2} \)
41 \( 1 + (1.06 - 1.84i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.77 + 10.0i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.885 + 1.53i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.92iT - 53T^{2} \)
59 \( 1 + (2.02 - 3.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.61 + 0.932i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.38 + 11.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.51iT - 71T^{2} \)
73 \( 1 + 1.90iT - 73T^{2} \)
79 \( 1 + (-0.433 - 0.751i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.45 + 5.99i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.77T + 89T^{2} \)
97 \( 1 + (0.200 - 0.115i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.831382405147135298027199262186, −8.162126832907972915101920046575, −7.47866207688141748811614004938, −6.62727037308989775913932416529, −6.33950302370755922259993387018, −4.73453827799801326269674298145, −3.82425910161830412684797265895, −2.81317615134105123934683936402, −2.22722916296479221725113774630, −0.40197861182711005146416132304, 1.51593318885007786136303585874, 2.74238253500859232292628170836, 4.01245326111996174434866824549, 4.34102507040643622764991537606, 5.26371907413306746451643691341, 6.27748574521421753426875979943, 7.53002575887805504513977438874, 8.261414787012349030113739247889, 8.558620053231688544402137855000, 9.750468986374113892345640378355

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