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Properties

Label 2-42e2-7.4-c3-0-24
Degree $2$
Conductor $1764$
Sign $0.947 - 0.318i$
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  + (5.32 + 9.22i)5-s + (−3.32 + 5.76i)11-s − 75.9·13-s + (52.1 − 90.3i)17-s + (42.7 + 74.0i)19-s + (−34.3 − 59.4i)23-s + (5.73 − 9.93i)25-s − 87.7·29-s + (31.3 − 54.3i)31-s + (−21.1 − 36.5i)37-s + 313.·41-s + 306.·43-s + (−107. − 186. i)47-s + (262. − 454. i)53-s − 70.8·55-s + ⋯
L(s)  = 1  + (0.476 + 0.825i)5-s + (−0.0911 + 0.157i)11-s − 1.62·13-s + (0.743 − 1.28i)17-s + (0.516 + 0.893i)19-s + (−0.311 − 0.539i)23-s + (0.0458 − 0.0794i)25-s − 0.562·29-s + (0.181 − 0.314i)31-s + (−0.0937 − 0.162i)37-s + 1.19·41-s + 1.08·43-s + (−0.333 − 0.578i)47-s + (0.680 − 1.17i)53-s − 0.173·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.947 - 0.318i$
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: \((2,\ 1764,\ (\ :3/2),\ 0.947 - 0.318i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.119688132\)
\(L(\frac12)\) \(\approx\) \(2.119688132\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.32 - 9.22i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (3.32 - 5.76i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 75.9T + 2.19e3T^{2} \)
17 \( 1 + (-52.1 + 90.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-42.7 - 74.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (34.3 + 59.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 87.7T + 2.43e4T^{2} \)
31 \( 1 + (-31.3 + 54.3i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (21.1 + 36.5i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 313.T + 6.89e4T^{2} \)
43 \( 1 - 306.T + 7.95e4T^{2} \)
47 \( 1 + (107. + 186. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-262. + 454. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (180. - 312. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-400. - 693. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-20.1 + 34.8i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 298.T + 3.57e5T^{2} \)
73 \( 1 + (-258. + 447. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-611. - 1.05e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.32e3T + 5.71e5T^{2} \)
89 \( 1 + (-319. - 554. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.42e3T + 9.12e5T^{2} \)
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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

−9.239654419157756574297121841467, −7.946392380124027592920370198542, −7.38533244156996679554315246801, −6.70917451753527191997015108219, −5.68611358952576684925779500224, −5.03922459306758777319510412588, −3.91936273952076580593949727757, −2.75440415045389573961537191009, −2.24229660194710335945736424820, −0.67292401630463636639007257009, 0.69100018902086055939703587386, 1.76741474626651733518356161344, 2.79747686989945340628727684155, 3.96279625334634946019045958075, 4.98812335279715450074751764379, 5.47204452546575430610409659862, 6.42460865586551631986856564026, 7.51028368094914901951990382179, 7.988666121792726036890376672771, 9.180391285348460051784470653212

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