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Properties

Label 2-3240-9.4-c1-0-7
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 + (−0.5 − 0.866i)11-s − 7·19-s + (−3 + 5.19i)23-s + (−0.499 − 0.866i)25-s + (3.5 + 6.06i)29-s + (−0.5 + 0.866i)31-s − 2·37-s + (−4.5 + 7.79i)41-s + (3 + 5.19i)43-s + (1 + 1.73i)47-s + (3.5 − 6.06i)49-s − 0.999·55-s + (−1.5 + 2.59i)59-s + (5 + 8.66i)61-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.150 − 0.261i)11-s − 1.60·19-s + (−0.625 + 1.08i)23-s + (−0.0999 − 0.173i)25-s + (0.649 + 1.12i)29-s + (−0.0898 + 0.155i)31-s − 0.328·37-s + (−0.702 + 1.21i)41-s + (0.457 + 0.792i)43-s + (0.145 + 0.252i)47-s + (0.5 − 0.866i)49-s − 0.134·55-s + (−0.195 + 0.338i)59-s + (0.640 + 1.10i)61-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

\(L(1)\) \(\approx\) \(0.9935386523\)
\(L(\frac12)\) \(\approx\) \(0.9935386523\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.5 - 6.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3 - 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
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 \( 1 - T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
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 \( 1 + 7T + 89T^{2} \)
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

−8.705550847400631241583998353425, −8.311867967741241590190328438237, −7.38221592576758449722141548656, −6.54794710929142880842536755335, −5.85927333511413202358990212732, −5.03952161400581239660405913048, −4.25481758404624996939780322094, −3.33419829508279863597697056492, −2.26887868781461605085451918475, −1.25285706293851046893731787580, 0.30101653910597770879682821785, 1.97793032664194551512798976238, 2.58169819546884810541221990767, 3.83428637614156016290375641886, 4.46591148130646932604413443463, 5.47316120474614583502508455091, 6.31727944437505973205507161240, 6.80291345804993444093805886201, 7.75607075380323243652641582952, 8.454310084063956440900752753337

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