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Properties

Label 2-54e2-9.7-c1-0-33
Degree $2$
Conductor $2916$
Sign $-1$
Analytic cond. $23.2843$
Root an. cond. $4.82538$
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  + (−1.22 − 2.11i)5-s + (2.23 − 3.86i)7-s + (−1.45 + 2.52i)11-s + (1.31 + 2.27i)13-s − 3.34·17-s − 2.04·19-s + (−4.62 − 8.01i)23-s + (−0.492 + 0.852i)25-s + (3.46 − 5.99i)29-s + (1.09 + 1.89i)31-s − 10.9·35-s − 1.41·37-s + (2.04 + 3.54i)41-s + (2.35 − 4.07i)43-s + (−2.99 + 5.18i)47-s + ⋯
L(s)  = 1  + (−0.547 − 0.947i)5-s + (0.843 − 1.46i)7-s + (−0.439 + 0.761i)11-s + (0.364 + 0.632i)13-s − 0.810·17-s − 0.469·19-s + (−0.964 − 1.67i)23-s + (−0.0984 + 0.170i)25-s + (0.643 − 1.11i)29-s + (0.196 + 0.340i)31-s − 1.84·35-s − 0.232·37-s + (0.319 + 0.553i)41-s + (0.359 − 0.621i)43-s + (−0.436 + 0.755i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2916\)    =    \(2^{2} \cdot 3^{6}\)
Sign: $-1$
Analytic conductor: \(23.2843\)
Root analytic conductor: \(4.82538\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2916} (1945, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2916,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8135767421\)
\(L(\frac12)\) \(\approx\) \(0.8135767421\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (1.22 + 2.11i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.23 + 3.86i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.45 - 2.52i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.31 - 2.27i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.34T + 17T^{2} \)
19 \( 1 + 2.04T + 19T^{2} \)
23 \( 1 + (4.62 + 8.01i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.46 + 5.99i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.09 - 1.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.41T + 37T^{2} \)
41 \( 1 + (-2.04 - 3.54i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.35 + 4.07i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.99 - 5.18i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.97T + 53T^{2} \)
59 \( 1 + (-3.30 - 5.71i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.61 + 6.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.55 + 2.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + 12.8T + 73T^{2} \)
79 \( 1 + (1.02 - 1.78i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.783 - 1.35i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.7T + 89T^{2} \)
97 \( 1 + (7.41 - 12.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.300500124784683848049933114894, −7.78984573280424163432962898828, −6.95335973610823894452886287167, −6.24064626151717272020660632640, −4.87925156448627437611559881600, −4.29618320460743081044695536855, −4.16866104854690677473745700571, −2.43794172875548484496221029101, −1.37335898953251059465771946976, −0.25698974499101017798682781234, 1.69041378958462286813919965594, 2.72113830014249690805334651209, 3.35343709803043060164248956187, 4.45705107857499621221511577810, 5.55740848531399959512688604271, 5.86267325867973929333636059461, 6.94077568831558487298090482456, 7.72470723868512206718213267396, 8.457715181498791162388837929012, 8.816800001906943564068161849767

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