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Properties

Label 2-42e2-7.5-c2-0-5
Degree $2$
Conductor $1764$
Sign $0.379 - 0.925i$
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  + (−3.68 − 2.12i)5-s + (−9.48 − 16.4i)11-s + 4.46i·13-s + (−11.0 + 6.38i)17-s + (8.11 + 4.68i)19-s + (−8.53 + 14.7i)23-s + (−3.44 − 5.97i)25-s − 20.8·29-s + (13.5 − 7.80i)31-s + (−22.8 + 39.5i)37-s − 63.3i·41-s + 2.20·43-s + (43.8 + 25.3i)47-s + (18.9 + 32.8i)53-s + 80.7i·55-s + ⋯
L(s)  = 1  + (−0.736 − 0.425i)5-s + (−0.862 − 1.49i)11-s + 0.343i·13-s + (−0.650 + 0.375i)17-s + (0.427 + 0.246i)19-s + (−0.371 + 0.642i)23-s + (−0.137 − 0.239i)25-s − 0.720·29-s + (0.435 − 0.251i)31-s + (−0.617 + 1.06i)37-s − 1.54i·41-s + 0.0511·43-s + (0.932 + 0.538i)47-s + (0.358 + 0.620i)53-s + 1.46i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.379 - 0.925i$
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: \((2,\ 1764,\ (\ :1),\ 0.379 - 0.925i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7719220001\)
\(L(\frac12)\) \(\approx\) \(0.7719220001\)
\(L(2)\) 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 \( 1 + (3.68 + 2.12i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (9.48 + 16.4i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 4.46iT - 169T^{2} \)
17 \( 1 + (11.0 - 6.38i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-8.11 - 4.68i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (8.53 - 14.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 20.8T + 841T^{2} \)
31 \( 1 + (-13.5 + 7.80i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (22.8 - 39.5i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 63.3iT - 1.68e3T^{2} \)
43 \( 1 - 2.20T + 1.84e3T^{2} \)
47 \( 1 + (-43.8 - 25.3i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-18.9 - 32.8i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-58.5 + 33.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-54.8 - 31.6i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (13.6 + 23.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 15.1T + 5.04e3T^{2} \)
73 \( 1 + (-72.5 + 41.9i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (66.5 - 115. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 109. iT - 6.88e3T^{2} \)
89 \( 1 + (69.2 + 39.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 119. iT - 9.40e3T^{2} \)
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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

−9.069203446174384222322502831368, −8.384773211626496490973343944141, −7.891623470169789027048814615939, −6.96018259598244152826091709235, −5.90660026227244554939492393860, −5.28422295236714024871359707124, −4.15377597237646264606849889313, −3.48153000301243774373691960618, −2.31584874145436471437649969404, −0.842028956300217045699322320526, 0.26271714759700679504529998547, 1.99733289569093566713678219165, 2.89636787540438107341436616244, 4.00089910214514978212638251946, 4.78124421237322331545823237515, 5.63381604456498367909203944552, 6.88970454192292049574356484691, 7.31945156735106520620136589090, 8.027362562440435674596664594799, 8.939123653709014201812613883412

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