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Properties

Label 2-42e2-84.59-c0-0-7
Degree $2$
Conductor $1764$
Sign $0.825 + 0.564i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.382 + 0.662i)5-s − 0.999i·8-s + (0.662 + 0.382i)10-s + 1.84i·13-s + (−0.5 − 0.866i)16-s + (0.923 − 1.60i)17-s + 0.765·20-s + (0.207 − 0.358i)25-s + (0.923 + 1.60i)26-s + (−0.866 − 0.499i)32-s − 1.84i·34-s + (0.707 + 1.22i)37-s + (0.662 − 0.382i)40-s − 1.84·41-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.382 + 0.662i)5-s − 0.999i·8-s + (0.662 + 0.382i)10-s + 1.84i·13-s + (−0.5 − 0.866i)16-s + (0.923 − 1.60i)17-s + 0.765·20-s + (0.207 − 0.358i)25-s + (0.923 + 1.60i)26-s + (−0.866 − 0.499i)32-s − 1.84i·34-s + (0.707 + 1.22i)37-s + (0.662 − 0.382i)40-s − 1.84·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.825 + 0.564i$
Analytic conductor: \(0.880350\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ 0.825 + 0.564i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.013805128\)
\(L(\frac12)\) \(\approx\) \(2.013805128\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - 1.84iT - T^{2} \)
17 \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + 1.84T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.662 + 0.382i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + 0.765iT - T^{2} \)
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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

−9.721464841424255399575140768446, −8.881951754620722377940858906124, −7.53548853058130414020265449473, −6.73310155523900688782875386238, −6.30275994475060995829430362818, −5.13602591006207575808147379749, −4.51373205123538664332291581556, −3.38191917105564735799613208797, −2.59676377532884464335547182099, −1.54072696811007900883727946681, 1.55757806277339257792812508640, 2.96874544329148399370901995574, 3.73092230198321249517509512253, 4.82623379601899489687377971304, 5.62964805944026581732057926832, 5.97742992783511721615481533330, 7.15905430656907688171886385373, 8.068707146547658814847067684186, 8.381022781560719244214749004806, 9.509827435564306336452136260230

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