Properties

Label 2-3240-9.4-c1-0-32
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 + (1.18 + 2.05i)7-s + (−1.68 − 2.92i)11-s + (−1.18 + 2.05i)13-s − 4.74·17-s + 19-s + (−0.186 + 0.322i)23-s + (−0.499 − 0.866i)25-s + (−1.68 − 2.92i)29-s + (3.05 − 5.29i)31-s + 2.37·35-s + 6·37-s + (5.87 − 10.1i)41-s + (−3.37 − 5.84i)43-s + (−1.81 − 3.14i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (0.448 + 0.776i)7-s + (−0.508 − 0.880i)11-s + (−0.328 + 0.569i)13-s − 1.15·17-s + 0.229·19-s + (−0.0388 + 0.0672i)23-s + (−0.0999 − 0.173i)25-s + (−0.313 − 0.542i)29-s + (0.549 − 0.951i)31-s + 0.400·35-s + 0.986·37-s + (0.917 − 1.58i)41-s + (−0.514 − 0.890i)43-s + (−0.264 − 0.458i)47-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: $\chi_{3240} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.454197669\)
\(L(\frac12)\) \(\approx\) \(1.454197669\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.18 - 2.05i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.68 + 2.92i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.18 - 2.05i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.74T + 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + (0.186 - 0.322i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.68 + 2.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.05 + 5.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + (-5.87 + 10.1i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.37 + 5.84i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.81 + 3.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.11T + 53T^{2} \)
59 \( 1 + (-2.5 + 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.627 + 1.08i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.37 - 9.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.37T + 71T^{2} \)
73 \( 1 - 3.25T + 73T^{2} \)
79 \( 1 + (4.37 + 7.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5 + 8.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.37T + 89T^{2} \)
97 \( 1 + (3.37 + 5.84i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.665465751586427530188620630444, −7.85372099658644272220522693457, −7.02019886011412688978816310250, −6.02252127372296267759680190717, −5.54105862305303005650400098581, −4.67692977067778736597453657844, −3.87908493669760562453600206740, −2.57703439589641857720868267979, −2.00340359272702501072677485001, −0.46400530550641295319183729903, 1.17321888833871824271831428920, 2.33345830853821436962046102734, 3.13493548238853908723297514829, 4.40710131673675750102138561264, 4.76410206688183916965380006555, 5.83353693111733644051869985713, 6.70422370775092627075043268479, 7.35974477052713977669017572903, 7.917670739725615947470090316414, 8.773071225898285748457803507611

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