Properties

Label 2-42e2-49.5-c0-0-0
Degree $2$
Conductor $1764$
Sign $0.969 + 0.243i$
Analytic cond. $0.880350$
Root an. cond. $0.938270$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.988 − 0.149i)7-s + (0.880 − 0.702i)13-s + (−0.258 − 0.149i)19-s + (−0.988 − 0.149i)25-s + (−0.510 + 0.294i)31-s + (1.40 + 0.432i)37-s + (0.658 + 0.317i)43-s + (0.955 − 0.294i)49-s + (0.574 − 1.86i)61-s + (−0.0747 − 0.129i)67-s + (−0.202 + 1.34i)73-s + (−0.955 + 1.65i)79-s + (0.766 − 0.825i)91-s − 1.56i·97-s + (−1.85 − 0.139i)103-s + ⋯
L(s)  = 1  + (0.988 − 0.149i)7-s + (0.880 − 0.702i)13-s + (−0.258 − 0.149i)19-s + (−0.988 − 0.149i)25-s + (−0.510 + 0.294i)31-s + (1.40 + 0.432i)37-s + (0.658 + 0.317i)43-s + (0.955 − 0.294i)49-s + (0.574 − 1.86i)61-s + (−0.0747 − 0.129i)67-s + (−0.202 + 1.34i)73-s + (−0.955 + 1.65i)79-s + (0.766 − 0.825i)91-s − 1.56i·97-s + (−1.85 − 0.139i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.969 + 0.243i$
Analytic conductor: \(0.880350\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ 0.969 + 0.243i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.293317238\)
\(L(\frac12)\) \(\approx\) \(1.293317238\)
\(L(1)\) 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 \( 1 + (-0.988 + 0.149i)T \)
good5 \( 1 + (0.988 + 0.149i)T^{2} \)
11 \( 1 + (-0.733 - 0.680i)T^{2} \)
13 \( 1 + (-0.880 + 0.702i)T + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.0747 + 0.997i)T^{2} \)
19 \( 1 + (0.258 + 0.149i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.0747 + 0.997i)T^{2} \)
29 \( 1 + (-0.900 + 0.433i)T^{2} \)
31 \( 1 + (0.510 - 0.294i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1.40 - 0.432i)T + (0.826 + 0.563i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (-0.658 - 0.317i)T + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (-0.955 + 0.294i)T^{2} \)
53 \( 1 + (0.826 - 0.563i)T^{2} \)
59 \( 1 + (0.988 - 0.149i)T^{2} \)
61 \( 1 + (-0.574 + 1.86i)T + (-0.826 - 0.563i)T^{2} \)
67 \( 1 + (0.0747 + 0.129i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.202 - 1.34i)T + (-0.955 - 0.294i)T^{2} \)
79 \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.733 - 0.680i)T^{2} \)
97 \( 1 + 1.56iT - T^{2} \)
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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

−9.472155240863141165109132029862, −8.449941961159471632655182516293, −8.045063231168474062046732171404, −7.19339392090971013915631400051, −6.14968883095960434114865146671, −5.44146307880557167735088509523, −4.48262829426868045580289846506, −3.65569597363906581748545626953, −2.42610432357003008650800456132, −1.22579082936470773715311315130, 1.43144871964847281885777039486, 2.42304040440377002549256069704, 3.83741175307538339416856120707, 4.45597501185927854456364169275, 5.57029102228211600360147402825, 6.18519634096975817147301112899, 7.29322980979943306206455952451, 7.944434079479631561771895277033, 8.759995930752658773985104384435, 9.355121163035715106159764066293

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