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Properties

Label 2-42e2-21.11-c2-0-0
Degree $2$
Conductor $1764$
Sign $-0.986 - 0.160i$
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  + (2.01 − 1.16i)5-s + (7.70 + 4.44i)11-s − 8.58·13-s + (−17.4 − 10.0i)17-s + (2.93 + 5.08i)19-s + (−15.0 + 8.69i)23-s + (−9.79 + 16.9i)25-s − 19.0i·29-s + (2.35 − 4.07i)31-s + (−19.8 − 34.4i)37-s + 6.98i·41-s − 35.7·43-s + (−63.5 + 36.6i)47-s + (40.4 + 23.3i)53-s + 20.7·55-s + ⋯
L(s)  = 1  + (0.403 − 0.232i)5-s + (0.700 + 0.404i)11-s − 0.660·13-s + (−1.02 − 0.591i)17-s + (0.154 + 0.267i)19-s + (−0.654 + 0.377i)23-s + (−0.391 + 0.678i)25-s − 0.656i·29-s + (0.0759 − 0.131i)31-s + (−0.537 − 0.930i)37-s + 0.170i·41-s − 0.831·43-s + (−1.35 + 0.780i)47-s + (0.762 + 0.440i)53-s + 0.376·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.160i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.986 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.986 - 0.160i$
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: \((2,\ 1764,\ (\ :1),\ -0.986 - 0.160i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.09722569905\)
\(L(\frac12)\) \(\approx\) \(0.09722569905\)
\(L(2)\) 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 \( 1 + (-2.01 + 1.16i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-7.70 - 4.44i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 8.58T + 169T^{2} \)
17 \( 1 + (17.4 + 10.0i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-2.93 - 5.08i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (15.0 - 8.69i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 19.0iT - 841T^{2} \)
31 \( 1 + (-2.35 + 4.07i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (19.8 + 34.4i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 6.98iT - 1.68e3T^{2} \)
43 \( 1 + 35.7T + 1.84e3T^{2} \)
47 \( 1 + (63.5 - 36.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-40.4 - 23.3i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-20.8 - 12.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (36.6 + 63.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-16.0 + 27.7i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 114. iT - 5.04e3T^{2} \)
73 \( 1 + (-20.6 + 35.7i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-13.3 - 23.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 136. iT - 6.88e3T^{2} \)
89 \( 1 + (126. - 72.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 65.9T + 9.40e3T^{2} \)
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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

−9.564911612779890057986829722209, −8.841379590842709080254574759100, −7.86788773400279342264571788070, −7.09334443443153779947589790624, −6.32306682917354043403932456351, −5.41046667347304526145846646708, −4.58533998037642650264686625284, −3.69802606827863488797441943710, −2.42843282234642482022492467373, −1.53477721793857067283798187901, 0.02353551905975386068164230444, 1.59199838621251135683914554809, 2.56551428214542547093369505818, 3.67517097963199837578949898298, 4.58839849940109698640979435123, 5.52165464934669752838857367276, 6.53347864224760699799574646242, 6.85830001222119534760602195640, 8.132142338760288606632447354125, 8.676289130667277603978805892474

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