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Properties

Label 2-42e2-49.3-c0-0-0
Degree $2$
Conductor $1764$
Sign $-0.274 - 0.961i$
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  + (−0.365 + 0.930i)7-s + (−1.55 + 1.24i)13-s + (−1.61 + 0.930i)19-s + (0.365 + 0.930i)25-s + (1.17 + 0.680i)31-s + (1.40 − 1.29i)37-s + (−1.78 − 0.858i)43-s + (−0.733 − 0.680i)49-s + (1.32 + 1.42i)61-s + (−0.826 + 1.43i)67-s + (0.548 − 0.215i)73-s + (0.733 + 1.26i)79-s + (−0.587 − 1.90i)91-s − 1.56i·97-s + (−0.167 + 0.246i)103-s + ⋯
L(s)  = 1  + (−0.365 + 0.930i)7-s + (−1.55 + 1.24i)13-s + (−1.61 + 0.930i)19-s + (0.365 + 0.930i)25-s + (1.17 + 0.680i)31-s + (1.40 − 1.29i)37-s + (−1.78 − 0.858i)43-s + (−0.733 − 0.680i)49-s + (1.32 + 1.42i)61-s + (−0.826 + 1.43i)67-s + (0.548 − 0.215i)73-s + (0.733 + 1.26i)79-s + (−0.587 − 1.90i)91-s − 1.56i·97-s + (−0.167 + 0.246i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.274 - 0.961i$
Analytic conductor: \(0.880350\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ -0.274 - 0.961i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7653237019\)
\(L(\frac12)\) \(\approx\) \(0.7653237019\)
\(L(1)\) 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 \( 1 + (0.365 - 0.930i)T \)
good5 \( 1 + (-0.365 - 0.930i)T^{2} \)
11 \( 1 + (0.955 - 0.294i)T^{2} \)
13 \( 1 + (1.55 - 1.24i)T + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.826 - 0.563i)T^{2} \)
19 \( 1 + (1.61 - 0.930i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.826 - 0.563i)T^{2} \)
29 \( 1 + (-0.900 + 0.433i)T^{2} \)
31 \( 1 + (-1.17 - 0.680i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.40 + 1.29i)T + (0.0747 - 0.997i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (1.78 + 0.858i)T + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (0.733 + 0.680i)T^{2} \)
53 \( 1 + (0.0747 + 0.997i)T^{2} \)
59 \( 1 + (-0.365 + 0.930i)T^{2} \)
61 \( 1 + (-1.32 - 1.42i)T + (-0.0747 + 0.997i)T^{2} \)
67 \( 1 + (0.826 - 1.43i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (-0.548 + 0.215i)T + (0.733 - 0.680i)T^{2} \)
79 \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (-0.955 - 0.294i)T^{2} \)
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

−9.714490521548968212166206024888, −8.913597174780715938940791415255, −8.326852409993621433550167241234, −7.21204769002994785989324705132, −6.58917569887027356563718296683, −5.70969505752996765975276929971, −4.81488161385498421540891329398, −3.95875732969036900714205734194, −2.66999265226807694274770003678, −1.92719683531556375317729741188, 0.54408882113953902598296067263, 2.36801178250083128416213148718, 3.18421111788138313269918125658, 4.49825228991454747589270588122, 4.88491363054085725762209691712, 6.31599121857812702925562037982, 6.73719411044172293462028141027, 7.84914540446961035808838310766, 8.206417038410405124296802332101, 9.461734841124064988551166785497

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