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Properties

Label 2-3240-9.4-c1-0-46
Degree $2$
Conductor $3240$
Sign $-0.939 - 0.342i$
Analytic cond. $25.8715$
Root an. cond. $5.08640$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)5-s + (−1 − 1.73i)7-s + (2 + 3.46i)11-s + (1 − 1.73i)13-s − 5·17-s − 5·19-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + (−1 − 1.73i)29-s + (−3.5 + 6.06i)31-s − 1.99·35-s − 6·37-s + (−2 − 3.46i)43-s + (2 + 3.46i)47-s + (1.50 − 2.59i)49-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.377 − 0.654i)7-s + (0.603 + 1.04i)11-s + (0.277 − 0.480i)13-s − 1.21·17-s − 1.14·19-s + (0.104 − 0.180i)23-s + (−0.0999 − 0.173i)25-s + (−0.185 − 0.321i)29-s + (−0.628 + 1.08i)31-s − 0.338·35-s − 0.986·37-s + (−0.304 − 0.528i)43-s + (0.291 + 0.505i)47-s + (0.214 − 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $-0.939 - 0.342i$
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: \(1\)
Selberg data: \((2,\ 3240,\ (\ :1/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2 - 3.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (-7 + 12.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.5 - 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 12T + 73T^{2} \)
79 \( 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (8 + 13.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.481685264826619326907952318483, −7.26938505773011698913794415544, −6.84404953838356760133599024243, −6.09068094246640675306117823439, −5.05142703048046376322261375222, −4.31741149185476075733361302176, −3.65327248779224623622391518193, −2.37047146682838022570745201339, −1.45319610744050266347252752361, 0, 1.70125352708718831020417605578, 2.58833484462354392581394954370, 3.55759338673608571972482624234, 4.31675408822436863934957868959, 5.42422349853287108314339035297, 6.29118350891873887849157861984, 6.51370929533518965043793197687, 7.53529207026854980876950084569, 8.652771486281700802270853949903

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