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Properties

Label 2-2646-9.7-c1-0-5
Degree $2$
Conductor $2646$
Sign $0.422 - 0.906i$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.517 + 0.896i)5-s − 0.999·8-s + 1.03·10-s + (−0.133 + 0.232i)11-s + (0.896 + 1.55i)13-s + (−0.5 + 0.866i)16-s − 6.83·17-s − 4.38·19-s + (0.517 − 0.896i)20-s + (0.133 + 0.232i)22-s + (2.73 + 4.73i)23-s + (1.96 − 3.40i)25-s + 1.79·26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.231 + 0.400i)5-s − 0.353·8-s + 0.327·10-s + (−0.0403 + 0.0699i)11-s + (0.248 + 0.430i)13-s + (−0.125 + 0.216i)16-s − 1.65·17-s − 1.00·19-s + (0.115 − 0.200i)20-s + (0.0285 + 0.0494i)22-s + (0.569 + 0.986i)23-s + (0.392 − 0.680i)25-s + 0.351·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.422 - 0.906i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.422 - 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.333433615\)
\(L(\frac12)\) \(\approx\) \(1.333433615\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.517 - 0.896i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.133 - 0.232i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.896 - 1.55i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.83T + 17T^{2} \)
19 \( 1 + 4.38T + 19T^{2} \)
23 \( 1 + (-2.73 - 4.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.34 - 5.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.46T + 37T^{2} \)
41 \( 1 + (-4.31 - 7.46i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.133 - 0.232i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.378 - 0.656i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 10.9T + 53T^{2} \)
59 \( 1 + (-0.637 - 1.10i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.31 - 10.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.23 + 10.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.46T + 71T^{2} \)
73 \( 1 + 5.41T + 73T^{2} \)
79 \( 1 + (4.46 - 7.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.29 - 5.70i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.07T + 89T^{2} \)
97 \( 1 + (9.07 - 15.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.069430898785647362337613279030, −8.474695590631897327891703967657, −7.35836014521929690482811297822, −6.49801076311127886739193809480, −6.05063969197329179570555368951, −4.75883382212447270263802171926, −4.35275762374032254639276554214, −3.17621336617359618351002460851, −2.40628988757595518830022885550, −1.38777777582587973401121325985, 0.37608286951440586019149715419, 2.03859035898900920146664470968, 3.01828328418148232123461006968, 4.35685619960537945574319479058, 4.58811372933363090468088979615, 5.82304341513114286414476789448, 6.26752903400932846684288603062, 7.12927353924239498033796085633, 7.950120646666905359748944394193, 8.760611698349618937367105282170

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