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Properties

Label 2-3240-9.4-c1-0-30
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

\(L(1)\) \(\approx\) \(1.121855837\)
\(L(\frac12)\) \(\approx\) \(1.121855837\)
\(L(\frac{3}{2})\) 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

−8.546427124431478521912065956665, −7.72107542061064072730207554392, −6.94626203376860517157058496202, −6.34748410794748119229715275041, −5.50778085643179359860210389790, −4.43388319362375098944391287347, −3.91854136012179800051379803532, −2.73753060388801700091898098854, −1.95948385673889945936172138544, −0.37161786521798621901467683054, 1.11717829738032397418185346735, 2.22644791967771007060631995971, 3.37042191233296151122071031471, 4.13311091801798342121841708923, 4.98969988818598053042667425669, 5.77059179378850071528799696860, 6.58573833081599472373357771697, 7.37386854581611387259722473585, 8.110719199251530693463066107274, 8.886969200731195783610123488199

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