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Properties

Label 2-42e2-21.11-c2-0-16
Degree $2$
Conductor $1764$
Sign $0.976 - 0.216i$
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  + (6.60 − 3.81i)5-s + (4.41 + 2.55i)11-s + 20.7·13-s + (22.7 + 13.1i)17-s + (11.1 + 19.3i)19-s + (−34.4 + 19.8i)23-s + (16.5 − 28.6i)25-s − 7.62i·29-s + (−5.89 + 10.2i)31-s + (30.1 + 52.2i)37-s + 11.8i·41-s − 30.3·43-s + (−33.0 + 19.0i)47-s + (5.11 + 2.95i)53-s + 38.9·55-s + ⋯
L(s)  = 1  + (1.32 − 0.762i)5-s + (0.401 + 0.231i)11-s + 1.59·13-s + (1.33 + 0.772i)17-s + (0.588 + 1.01i)19-s + (−1.49 + 0.864i)23-s + (0.662 − 1.14i)25-s − 0.262i·29-s + (−0.190 + 0.329i)31-s + (0.815 + 1.41i)37-s + 0.287i·41-s − 0.705·43-s + (−0.703 + 0.406i)47-s + (0.0964 + 0.0556i)53-s + 0.707·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.976 - 0.216i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ 0.976 - 0.216i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.114124209\)
\(L(\frac12)\) \(\approx\) \(3.114124209\)
\(L(2)\) 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 \( 1 + (-6.60 + 3.81i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-4.41 - 2.55i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 20.7T + 169T^{2} \)
17 \( 1 + (-22.7 - 13.1i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-11.1 - 19.3i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (34.4 - 19.8i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 7.62iT - 841T^{2} \)
31 \( 1 + (5.89 - 10.2i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-30.1 - 52.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 11.8iT - 1.68e3T^{2} \)
43 \( 1 + 30.3T + 1.84e3T^{2} \)
47 \( 1 + (33.0 - 19.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-5.11 - 2.95i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-38.2 - 22.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (24.7 + 42.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (31.5 - 54.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 41.6iT - 5.04e3T^{2} \)
73 \( 1 + (2.08 - 3.61i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-20.8 - 36.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 145. iT - 6.88e3T^{2} \)
89 \( 1 + (38.0 - 21.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 39.5T + 9.40e3T^{2} \)
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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

−9.243476113584040690948391665263, −8.328252976329618151223627197854, −7.83716473528444745473967020613, −6.36149450661775312160999996718, −5.94471415538512272210133712137, −5.31329907679656931799453310610, −4.09559364494262433067402234510, −3.26728995801569193413241081049, −1.59862380903962026508740446226, −1.36084919712681912641688969989, 0.908588184103309211267015565575, 2.05438406090068790183962572040, 3.03400200695606223979205688692, 3.90898869284476824952253248575, 5.27116074180379929915857122484, 5.96487890073195691145220835203, 6.50164583642379286433618692967, 7.40289286928918002170281779656, 8.384912934905969364217702487312, 9.242399476362357059757761714415

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