Properties

Label 2-42e2-21.17-c3-0-19
Degree $2$
Conductor $1764$
Sign $0.154 - 0.987i$
Analytic cond. $104.079$
Root an. cond. $10.2019$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (10.1 + 17.5i)5-s + (−15.4 − 8.92i)11-s − 33.1i·13-s + (22.9 − 39.6i)17-s + (35.0 − 20.2i)19-s + (−69.7 + 40.2i)23-s + (−142. + 246. i)25-s − 233. i·29-s + (195. + 112. i)31-s + (135. + 234. i)37-s + 154.·41-s + 367.·43-s + (263. + 457. i)47-s + (78.8 + 45.5i)53-s − 361. i·55-s + ⋯
L(s)  = 1  + (0.905 + 1.56i)5-s + (−0.423 − 0.244i)11-s − 0.707i·13-s + (0.326 − 0.566i)17-s + (0.422 − 0.244i)19-s + (−0.632 + 0.365i)23-s + (−1.14 + 1.97i)25-s − 1.49i·29-s + (1.13 + 0.653i)31-s + (0.601 + 1.04i)37-s + 0.588·41-s + 1.30·43-s + (0.818 + 1.41i)47-s + (0.204 + 0.118i)53-s − 0.885i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.154 - 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.154 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.154 - 0.987i$
Analytic conductor: \(104.079\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ 0.154 - 0.987i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.536132103\)
\(L(\frac12)\) \(\approx\) \(2.536132103\)
\(L(\frac{5}{2})\) 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 \( 1 + (-10.1 - 17.5i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (15.4 + 8.92i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 33.1iT - 2.19e3T^{2} \)
17 \( 1 + (-22.9 + 39.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-35.0 + 20.2i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (69.7 - 40.2i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 233. iT - 2.43e4T^{2} \)
31 \( 1 + (-195. - 112. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-135. - 234. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 154.T + 6.89e4T^{2} \)
43 \( 1 - 367.T + 7.95e4T^{2} \)
47 \( 1 + (-263. - 457. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-78.8 - 45.5i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-312. + 541. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-78.8 + 45.5i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (431. - 747. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 303. iT - 3.57e5T^{2} \)
73 \( 1 + (-999. - 576. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-3.48 - 6.02i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 815.T + 5.71e5T^{2} \)
89 \( 1 + (155. + 269. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.83e3iT - 9.12e5T^{2} \)
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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

−9.442355623211761731480867593449, −8.097058001854701031319466925491, −7.54181718474328664361767090785, −6.63256463464060364224410818000, −5.97190317804897598178689489527, −5.30669256036148982807969005889, −3.98146391503147429171317194408, −2.76311992329447726672086747905, −2.58484392660914460287492485303, −0.999201223581665633324522601299, 0.61039354121528913961778899173, 1.58242251634078591076344351054, 2.43154849231270377877042980022, 3.97798666593167869009155127639, 4.68452023859426951235276744819, 5.55961137318818211891696784128, 6.06787535854593264035217720196, 7.24376193477278766763130228555, 8.165040891729598469801618529212, 8.829095516320580420115139195911

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