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Properties

Label 2-42e2-7.2-c3-0-47
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  + (8.32 − 14.4i)5-s + (−35.8 − 62.1i)11-s + 65.3·13-s + (−45.2 − 78.3i)17-s + (81.9 − 141. i)19-s + (39.6 − 68.6i)23-s + (−76.1 − 131. i)25-s + 43.2·29-s + (67.8 + 117. i)31-s + (−135. + 234. i)37-s + 152.·41-s − 177.·43-s + (22.8 − 39.5i)47-s + (−79.2 − 137. i)53-s − 1.19e3·55-s + ⋯
L(s)  = 1  + (0.744 − 1.29i)5-s + (−0.983 − 1.70i)11-s + 1.39·13-s + (−0.645 − 1.11i)17-s + (0.989 − 1.71i)19-s + (0.359 − 0.622i)23-s + (−0.609 − 1.05i)25-s + 0.277·29-s + (0.392 + 0.680i)31-s + (−0.601 + 1.04i)37-s + 0.579·41-s − 0.630·43-s + (0.0708 − 0.122i)47-s + (−0.205 − 0.355i)53-s − 2.93·55-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

\(L(2)\) \(\approx\) \(2.396637069\)
\(L(\frac12)\) \(\approx\) \(2.396637069\)
\(L(\frac{5}{2})\) 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 \( 1 + (-8.32 + 14.4i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (35.8 + 62.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 65.3T + 2.19e3T^{2} \)
17 \( 1 + (45.2 + 78.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-81.9 + 141. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-39.6 + 68.6i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 43.2T + 2.43e4T^{2} \)
31 \( 1 + (-67.8 - 117. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (135. - 234. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 152.T + 6.89e4T^{2} \)
43 \( 1 + 177.T + 7.95e4T^{2} \)
47 \( 1 + (-22.8 + 39.5i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (79.2 + 137. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-195. - 339. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-275. + 477. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (229. + 397. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 486.T + 3.57e5T^{2} \)
73 \( 1 + (-287. - 497. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-334. + 578. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 76.2T + 5.71e5T^{2} \)
89 \( 1 + (683. - 1.18e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 242.T + 9.12e5T^{2} \)
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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

−8.711515946617368848284092813001, −8.155409794265827714652161864934, −6.87179728202739378112796930087, −6.08165825421533488208767306558, −5.14575114693431456198446453538, −4.87371989059854072415944435530, −3.37430947090744999009211927348, −2.56164683230250128621251687191, −1.06102513046464211457337966011, −0.55789240474903158975653750166, 1.55437087984917348918888907581, 2.24023447967234734150769185113, 3.33588708107699080915692894774, 4.17134539914322392419670630180, 5.48161957508674115040743900842, 6.05047192521871448633918206274, 6.89328197539328353295116220359, 7.59608569495487152775456952984, 8.372494616504279968180372741172, 9.550027745940560039856825295432

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