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Properties

Label 2-42e2-7.4-c3-0-36
Degree $2$
Conductor $1764$
Sign $-0.386 + 0.922i$
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  + (−4.91 − 8.50i)5-s + (−7.08 + 12.2i)11-s + 26.1·13-s + (39.2 − 68.0i)17-s + (36.5 + 63.3i)19-s + (−48 − 83.1i)23-s + (14.2 − 24.6i)25-s − 173.·29-s + (33.6 − 58.2i)31-s + (150. + 261. i)37-s + 472.·41-s − 463.·43-s + (45.5 + 78.9i)47-s + (−81.6 + 141. i)53-s + 139.·55-s + ⋯
L(s)  = 1  + (−0.439 − 0.761i)5-s + (−0.194 + 0.336i)11-s + 0.557·13-s + (0.560 − 0.971i)17-s + (0.441 + 0.765i)19-s + (−0.435 − 0.753i)23-s + (0.113 − 0.197i)25-s − 1.10·29-s + (0.194 − 0.337i)31-s + (0.670 + 1.16i)37-s + 1.79·41-s − 1.64·43-s + (0.141 + 0.245i)47-s + (−0.211 + 0.366i)53-s + 0.341·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.386 + 0.922i$
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: \((2,\ 1764,\ (\ :3/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.470318080\)
\(L(\frac12)\) \(\approx\) \(1.470318080\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.91 + 8.50i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (7.08 - 12.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 26.1T + 2.19e3T^{2} \)
17 \( 1 + (-39.2 + 68.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-36.5 - 63.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (48 + 83.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 173.T + 2.43e4T^{2} \)
31 \( 1 + (-33.6 + 58.2i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-150. - 261. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 472.T + 6.89e4T^{2} \)
43 \( 1 + 463.T + 7.95e4T^{2} \)
47 \( 1 + (-45.5 - 78.9i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (81.6 - 141. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-300. + 520. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-285. - 495. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-269. + 467. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 1.06e3T + 3.57e5T^{2} \)
73 \( 1 + (221. - 383. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-22.8 - 39.5i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 686.T + 5.71e5T^{2} \)
89 \( 1 + (330. + 571. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 658.T + 9.12e5T^{2} \)
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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

−8.523680936659320222347906395443, −8.027888745297749925820426604158, −7.23996826840656797349560237418, −6.22142935012305860278703115005, −5.36282049026696036450334420513, −4.55741743532300451040054204209, −3.74486097139657490525757977328, −2.64284819451083281110859876835, −1.36091262106090480720099802538, −0.36880183843912319308806946224, 1.02707383036764965985928428319, 2.31196528364723332290950577583, 3.42478008571960593359536111452, 3.91388056783221392123862158062, 5.26180707387809449980781221281, 5.96554482202900319894630159005, 6.88143475897255555070236219316, 7.61093121155795388775831177054, 8.269378615374766825606700924179, 9.210398047126454424103858797140

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