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Properties

Label 2-3240-9.4-c1-0-31
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: $\chi_{3240} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.655676292\)
\(L(\frac12)\) \(\approx\) \(1.655676292\)
\(L(\frac{3}{2})\) 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

−8.540223972768996793226948795706, −7.59762911680978690345969509947, −7.13283953037680506822708484828, −6.19879515558044651203530969823, −5.47748566771949301448685266056, −4.55833196703981417968026599152, −3.85022534545496910170789524940, −2.86739079588172408049261016459, −1.71498687479888353872801320952, −0.56611576397386358411542041566, 1.14665309631874332067934706220, 2.44945930128364634824539687876, 3.13818904541381699451621627970, 4.03304808304314484724377918909, 5.25117483585517765213115052153, 5.71957074898778087340546010500, 6.55536425793309974024182979886, 7.31804891056379523800901404089, 8.030931578743356619696315012254, 8.978996037818273424428764250109

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