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Properties

Label 2-42e2-7.2-c3-0-44
Degree $2$
Conductor $1764$
Sign $-0.900 + 0.435i$
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  + (9.57 − 16.5i)5-s + (20.2 + 35.1i)11-s − 50.4·13-s + (25.9 + 44.9i)17-s + (−16.5 + 28.6i)19-s + (31.4 − 54.4i)23-s + (−120. − 209. i)25-s − 129.·29-s + (121. + 209. i)31-s + (194. − 337. i)37-s − 470.·41-s − 125.·43-s + (193. − 334. i)47-s + (−305. − 529. i)53-s + 777.·55-s + ⋯
L(s)  = 1  + (0.855 − 1.48i)5-s + (0.556 + 0.963i)11-s − 1.07·13-s + (0.370 + 0.641i)17-s + (−0.200 + 0.346i)19-s + (0.284 − 0.493i)23-s + (−0.965 − 1.67i)25-s − 0.831·29-s + (0.702 + 1.21i)31-s + (0.865 − 1.49i)37-s − 1.79·41-s − 0.443·43-s + (0.599 − 1.03i)47-s + (−0.792 − 1.37i)53-s + 1.90·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.900 + 0.435i$
Analytic conductor: \(104.079\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ -0.900 + 0.435i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.320921210\)
\(L(\frac12)\) \(\approx\) \(1.320921210\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.57 + 16.5i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-20.2 - 35.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 50.4T + 2.19e3T^{2} \)
17 \( 1 + (-25.9 - 44.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (16.5 - 28.6i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-31.4 + 54.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 129.T + 2.43e4T^{2} \)
31 \( 1 + (-121. - 209. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-194. + 337. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 470.T + 6.89e4T^{2} \)
43 \( 1 + 125.T + 7.95e4T^{2} \)
47 \( 1 + (-193. + 334. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (305. + 529. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (113. + 196. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-362. + 628. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (522. + 905. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 169.T + 3.57e5T^{2} \)
73 \( 1 + (190. + 330. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-580. + 1.00e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 808.T + 5.71e5T^{2} \)
89 \( 1 + (-159. + 277. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.13e3T + 9.12e5T^{2} \)
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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

−8.700843699993296457455359285134, −7.961937900505468289498599861850, −6.96055233334484590327336523927, −6.12002119744790880927581527731, −5.09006969509845943497966018840, −4.77550483090231157692113484066, −3.66240222753119112539830698024, −2.11072814043140903664677507751, −1.54998587337137920363213160202, −0.26041146189697737041048973756, 1.31892046860271278619219251326, 2.65736271381089980848041455754, 3.00184894855808617713939352094, 4.25353529008347226159613884278, 5.46928601552010581439136905263, 6.09927747289546857943847332577, 6.90390297637433131046787673186, 7.45800521037813138710540350167, 8.507511294735002314188679725268, 9.661638018334694425709011464916

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