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Properties

Label 2-42e2-21.2-c2-0-13
Degree $2$
Conductor $1764$
Sign $-0.529 + 0.848i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.529 + 0.848i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1745, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ -0.529 + 0.848i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9368803852\)
\(L(\frac12)\) \(\approx\) \(0.9368803852\)
\(L(2)\) 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

−9.011884190846710709039598132776, −8.041606372259690209825861459679, −7.44331730286135827188373093775, −6.51981284944834304426394599549, −5.49938571861716233766925320251, −4.53695907221896343879252992668, −3.95501412291822892382139107502, −2.96446058034307030404032261872, −1.42070951255444073721862219833, −0.30542596914294687256076729350, 1.05788605634157605063398891569, 2.73290863751949844274630042653, 3.45410596975797331862851422408, 4.26094064908295770186163038677, 5.24922392632589016547398320851, 6.47080194824410652417059534243, 6.91549587814501316117611063489, 7.893609423834843084256018277844, 8.449996261414035821667688009016, 9.204924651640818875929238626152

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