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Properties

Label 2-1620-9.4-c1-0-5
Degree $2$
Conductor $1620$
Sign $0.766 - 0.642i$
Analytic cond. $12.9357$
Root an. cond. $3.59663$
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)5-s + (2 + 3.46i)7-s + (−1.5 − 2.59i)11-s + (2 − 3.46i)13-s + 5·19-s + (3 − 5.19i)23-s + (−0.499 − 0.866i)25-s + (4.5 + 7.79i)29-s + (−2.5 + 4.33i)31-s − 3.99·35-s + 2·37-s + (4.5 − 7.79i)41-s + (5 + 8.66i)43-s + (3 + 5.19i)47-s + (−4.49 + 7.79i)49-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (0.755 + 1.30i)7-s + (−0.452 − 0.783i)11-s + (0.554 − 0.960i)13-s + 1.14·19-s + (0.625 − 1.08i)23-s + (−0.0999 − 0.173i)25-s + (0.835 + 1.44i)29-s + (−0.449 + 0.777i)31-s − 0.676·35-s + 0.328·37-s + (0.702 − 1.21i)41-s + (0.762 + 1.32i)43-s + (0.437 + 0.757i)47-s + (−0.642 + 1.11i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.839905429\)
\(L(\frac12)\) \(\approx\) \(1.839905429\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3 + 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.248839847856189721985601763268, −8.638352095944642759694787610973, −8.035605832404872760247455687383, −7.19097435233571141372816542410, −6.02369427964775799546206805766, −5.48974204522978763605888325413, −4.65721685080376102509392771859, −3.17044565313194721402684349304, −2.71347237714298722036415788905, −1.14293069379841707352992722486, 0.895295443662450554357463963374, 1.96205772729715479995513566826, 3.50832141578682170466398750906, 4.37140213572430849281427805456, 4.92972423493345658179987439707, 6.06673908830539985331608599105, 7.21863006699407120818014829366, 7.58586718954105188407062594631, 8.364529587754795985425252141992, 9.509584145005958275689704830079

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