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Properties

Label 2-54e2-9.4-c1-0-20
Degree $2$
Conductor $2916$
Sign $0.5 - 0.866i$
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  + (−1.15 + 1.99i)5-s + (0.466 + 0.808i)7-s + (2.42 + 4.20i)11-s + (3.04 − 5.26i)13-s + 2.85·17-s + 5.01·19-s + (−0.991 + 1.71i)23-s + (−0.147 − 0.255i)25-s + (−0.939 − 1.62i)29-s + (−0.666 + 1.15i)31-s − 2.14·35-s − 6.00·37-s + (5.59 − 9.68i)41-s + (−3.78 − 6.55i)43-s + (5.46 + 9.46i)47-s + ⋯
L(s)  = 1  + (−0.514 + 0.891i)5-s + (0.176 + 0.305i)7-s + (0.731 + 1.26i)11-s + (0.843 − 1.46i)13-s + 0.693·17-s + 1.15·19-s + (−0.206 + 0.357i)23-s + (−0.0295 − 0.0511i)25-s + (−0.174 − 0.302i)29-s + (−0.119 + 0.207i)31-s − 0.363·35-s − 0.987·37-s + (0.873 − 1.51i)41-s + (−0.577 − 0.999i)43-s + (0.797 + 1.38i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2916\)    =    \(2^{2} \cdot 3^{6}\)
Sign: $0.5 - 0.866i$
Analytic conductor: \(23.2843\)
Root analytic conductor: \(4.82538\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2916} (973, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2916,\ (\ :1/2),\ 0.5 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.935885939\)
\(L(\frac12)\) \(\approx\) \(1.935885939\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.15 - 1.99i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.466 - 0.808i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.42 - 4.20i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.04 + 5.26i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.85T + 17T^{2} \)
19 \( 1 - 5.01T + 19T^{2} \)
23 \( 1 + (0.991 - 1.71i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.939 + 1.62i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.666 - 1.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.00T + 37T^{2} \)
41 \( 1 + (-5.59 + 9.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.78 + 6.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.46 - 9.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.60T + 53T^{2} \)
59 \( 1 + (-2.19 + 3.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.52 - 9.57i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.386 + 0.669i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.25T + 71T^{2} \)
73 \( 1 + 0.443T + 73T^{2} \)
79 \( 1 + (2.61 + 4.53i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.91 - 13.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 + (-3.48 - 6.04i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.888562841581131872558766983172, −7.943156266856422693693683844775, −7.39188580823559377740226960693, −6.81425851263855375782594645065, −5.72270071271660906887871077569, −5.19894821050322792375844001156, −3.85827958009667733438694142109, −3.42289803180976461207114332158, −2.34751581045848350228336032681, −1.08449860181659386307564143721, 0.798390993231997104520702312397, 1.55520362052061587040261942602, 3.19661125329181218674506741495, 3.91621654230395331434544991068, 4.60946850319385906011558025703, 5.57199643650803434467876590210, 6.33624060040370660228383544701, 7.13836499662019836751127771148, 8.091803315477797257150797432890, 8.603854383133353351152442792494

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