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Properties

Label 2-42e2-7.4-c3-0-46
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  + (−4.08 − 7.07i)5-s + (18.9 − 32.7i)11-s − 39.9·13-s + (−4.96 + 8.60i)17-s + (−45.2 − 78.3i)19-s + (59.2 + 102. i)23-s + (29.1 − 50.4i)25-s + 78.4·29-s + (46.0 − 79.6i)31-s + (−166. − 287. i)37-s + 71.7·41-s − 115.·43-s + (−153. − 266. i)47-s + (−201. + 349. i)53-s − 308.·55-s + ⋯
L(s)  = 1  + (−0.365 − 0.632i)5-s + (0.518 − 0.897i)11-s − 0.851·13-s + (−0.0708 + 0.122i)17-s + (−0.546 − 0.945i)19-s + (0.537 + 0.931i)23-s + (0.233 − 0.403i)25-s + 0.502·29-s + (0.266 − 0.461i)31-s + (−0.738 − 1.27i)37-s + 0.273·41-s − 0.411·43-s + (−0.477 − 0.827i)47-s + (−0.522 + 0.904i)53-s − 0.757·55-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: $\chi_{1764} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ -0.900 - 0.435i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4175162608\)
\(L(\frac12)\) \(\approx\) \(0.4175162608\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.08 + 7.07i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-18.9 + 32.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 39.9T + 2.19e3T^{2} \)
17 \( 1 + (4.96 - 8.60i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (45.2 + 78.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-59.2 - 102. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 78.4T + 2.43e4T^{2} \)
31 \( 1 + (-46.0 + 79.6i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (166. + 287. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 71.7T + 6.89e4T^{2} \)
43 \( 1 + 115.T + 7.95e4T^{2} \)
47 \( 1 + (153. + 266. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (201. - 349. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (296. - 514. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-166. - 288. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-371. + 643. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 728.T + 3.57e5T^{2} \)
73 \( 1 + (400. - 694. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (533. + 924. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 906.T + 5.71e5T^{2} \)
89 \( 1 + (556. + 964. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.48e3T + 9.12e5T^{2} \)
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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

−8.675607040162354789796154967604, −7.72200504263288516589015439728, −6.94840480551450337964268766517, −6.05535336772593491245177106765, −5.11344550285040177635984154066, −4.39417523843662081696661012433, −3.43149783214381359623786056950, −2.36775362887888247253353173848, −1.03947452980684761663588531216, −0.098021926270086367758345633106, 1.45863755029840856193735974967, 2.55252143697136128192107558559, 3.47730550322206580368180519381, 4.49103942773578246700163574319, 5.17173111108221024168857169028, 6.61467487745660961853910144136, 6.77058590407135457433880931944, 7.81328889464310844504058064363, 8.487479136699608680203578592767, 9.544876659005787121579952934273

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