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Properties

Label 2-3240-9.7-c1-0-10
Degree $2$
Conductor $3240$
Sign $0.173 - 0.984i$
Analytic cond. $25.8715$
Root an. cond. $5.08640$
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 + (−1 + 1.73i)7-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s − 17-s + 4·19-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + (2.5 − 4.33i)29-s + (−0.5 − 0.866i)31-s + 1.99·35-s + 6·37-s + (−3.5 + 6.06i)43-s + (−3.5 + 6.06i)47-s + (1.50 + 2.59i)49-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.377 + 0.654i)7-s + (−0.150 + 0.261i)11-s + (−0.138 − 0.240i)13-s − 0.242·17-s + 0.917·19-s + (−0.104 − 0.180i)23-s + (−0.0999 + 0.173i)25-s + (0.464 − 0.804i)29-s + (−0.0898 − 0.155i)31-s + 0.338·35-s + 0.986·37-s + (−0.533 + 0.924i)43-s + (−0.510 + 0.884i)47-s + (0.214 + 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(25.8715\)
Root analytic conductor: \(5.08640\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3240} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.198960450\)
\(L(\frac12)\) \(\approx\) \(1.198960450\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.5 + 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.5 - 6.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.5 - 6.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + (7.5 - 12.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1 - 1.73i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + (5 - 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.811180098204356398035596160162, −8.052977308860744675381493226106, −7.48206075104480378644952154314, −6.46194394731606185759539926355, −5.82933769369520018390197761801, −4.96802095221989021646959852615, −4.25734716646285136907031146503, −3.14841587861224694340556817538, −2.39830636531264772134802039585, −1.06921554210307548667931742432, 0.41774804912269872694536073101, 1.78444074223996047091363462196, 3.05153962687382360164458365677, 3.60704971094001983496791193509, 4.61181274697251050954562419488, 5.40671288368061186665201614622, 6.42973201773822391881252431866, 6.96029358128351868668549425402, 7.69273933037273209090264112468, 8.396147865566301539098797273283

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