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Properties

Label 2-42e2-7.5-c2-0-22
Degree $2$
Conductor $1764$
Sign $0.937 + 0.347i$
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  + (5.33 + 3.07i)5-s + (−0.975 − 1.69i)11-s − 0.317i·13-s + (15.9 − 9.23i)17-s + (−6.01 − 3.47i)19-s + (18.4 − 31.9i)23-s + (6.44 + 11.1i)25-s − 40.7·29-s + (26.2 − 15.1i)31-s + (6.84 − 11.8i)37-s − 30.1i·41-s + 41.7·43-s + (42.1 + 24.3i)47-s + (1.95 + 3.38i)53-s − 12.0i·55-s + ⋯
L(s)  = 1  + (1.06 + 0.615i)5-s + (−0.0887 − 0.153i)11-s − 0.0243i·13-s + (0.940 − 0.543i)17-s + (−0.316 − 0.182i)19-s + (0.801 − 1.38i)23-s + (0.257 + 0.446i)25-s − 1.40·29-s + (0.845 − 0.488i)31-s + (0.185 − 0.320i)37-s − 0.735i·41-s + 0.972·43-s + (0.896 + 0.517i)47-s + (0.0368 + 0.0637i)53-s − 0.218i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.937 + 0.347i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ 0.937 + 0.347i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.549864769\)
\(L(\frac12)\) \(\approx\) \(2.549864769\)
\(L(2)\) 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 \( 1 + (-5.33 - 3.07i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (0.975 + 1.69i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 0.317iT - 169T^{2} \)
17 \( 1 + (-15.9 + 9.23i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (6.01 + 3.47i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-18.4 + 31.9i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 40.7T + 841T^{2} \)
31 \( 1 + (-26.2 + 15.1i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-6.84 + 11.8i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 30.1iT - 1.68e3T^{2} \)
43 \( 1 - 41.7T + 1.84e3T^{2} \)
47 \( 1 + (-42.1 - 24.3i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-1.95 - 3.38i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-20.7 + 12.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (69.1 + 39.9i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-45.6 - 79.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 75.7T + 5.04e3T^{2} \)
73 \( 1 + (-114. + 66.0i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-12.5 + 21.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 84.9iT - 6.88e3T^{2} \)
89 \( 1 + (110. + 64.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 4.05iT - 9.40e3T^{2} \)
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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

−9.299474620846960597669926416565, −8.292055971783396612104116763022, −7.39545985530055004558342782977, −6.61169378239762403905298780011, −5.87630207286190992776542720597, −5.16187021093680584256085906489, −4.02536199668864655062916608178, −2.84865516769260137795329339336, −2.18472560445751988355693787269, −0.75934610540822007825974619690, 1.10211440781374723122658880146, 1.93192048069690662580774743953, 3.15746430893227259325133921140, 4.21826561601908612196161877052, 5.38429832828845002095327350218, 5.66170170544533647071347650473, 6.71009675369045931100536189295, 7.64617780880300696701555732333, 8.415057104189208798828963923075, 9.420991838830698235920574373356

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