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Properties

Label 2-42e2-7.5-c2-0-26
Degree $2$
Conductor $1764$
Sign $-0.580 + 0.814i$
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  + (−0.804 − 0.464i)5-s + (4.84 + 8.39i)11-s − 15.9i·13-s + (9.14 − 5.27i)17-s + (−6.25 − 3.61i)19-s + (−5.65 + 9.78i)23-s + (−12.0 − 20.9i)25-s − 46.3·29-s + (0.418 − 0.241i)31-s + (−1.24 + 2.14i)37-s + 55.8i·41-s + 60.6·43-s + (31.6 + 18.2i)47-s + (−14.2 − 24.7i)53-s − 9.00i·55-s + ⋯
L(s)  = 1  + (−0.160 − 0.0929i)5-s + (0.440 + 0.762i)11-s − 1.22i·13-s + (0.537 − 0.310i)17-s + (−0.329 − 0.190i)19-s + (−0.245 + 0.425i)23-s + (−0.482 − 0.836i)25-s − 1.59·29-s + (0.0134 − 0.00779i)31-s + (−0.0335 + 0.0580i)37-s + 1.36i·41-s + 1.41·43-s + (0.674 + 0.389i)47-s + (−0.269 − 0.466i)53-s − 0.163i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.580 + 0.814i$
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: \((2,\ 1764,\ (\ :1),\ -0.580 + 0.814i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9795475748\)
\(L(\frac12)\) \(\approx\) \(0.9795475748\)
\(L(2)\) 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 \( 1 + (0.804 + 0.464i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-4.84 - 8.39i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 15.9iT - 169T^{2} \)
17 \( 1 + (-9.14 + 5.27i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (6.25 + 3.61i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (5.65 - 9.78i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 46.3T + 841T^{2} \)
31 \( 1 + (-0.418 + 0.241i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (1.24 - 2.14i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 55.8iT - 1.68e3T^{2} \)
43 \( 1 - 60.6T + 1.84e3T^{2} \)
47 \( 1 + (-31.6 - 18.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (14.2 + 24.7i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-81.4 + 47.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (95.4 + 55.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (41.0 + 71.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 127.T + 5.04e3T^{2} \)
73 \( 1 + (-40.0 + 23.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (9.35 - 16.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 59.6iT - 6.88e3T^{2} \)
89 \( 1 + (61.5 + 35.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 102. iT - 9.40e3T^{2} \)
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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

−8.907976893172542811509834842433, −7.80075699075336049563750855999, −7.54378831555927084086886045931, −6.36035034424661533379905520871, −5.62567451926270926957494237058, −4.69026269730088749134244650864, −3.80720299446141687271274391290, −2.80413418764312222246839495699, −1.61746667838493583680456422307, −0.26248340041770569345538711694, 1.28704228395602373494050904377, 2.39028397700790036406175191752, 3.72211045234780522419539589220, 4.15974866215244282463788476051, 5.52841496616640892156303860915, 6.08269643757300407868715204020, 7.11600063183869891986609342074, 7.68992703061183503565871499971, 8.891452256566458969725393997005, 9.085675378867626056566143985601

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