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Properties

Label 2-54e2-9.7-c1-0-6
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  + (−0.497 − 0.861i)5-s + (0.719 − 1.24i)7-s + (−1.67 + 2.89i)11-s + (0.972 + 1.68i)13-s + 2.32·17-s − 6.77·19-s + (4.47 + 7.75i)23-s + (2.00 − 3.47i)25-s + (2.43 − 4.22i)29-s + (−3.09 − 5.36i)31-s − 1.43·35-s + 0.165·37-s + (5.39 + 9.34i)41-s + (−4.45 + 7.72i)43-s + (−3.27 + 5.66i)47-s + ⋯
L(s)  = 1  + (−0.222 − 0.385i)5-s + (0.271 − 0.471i)7-s + (−0.504 + 0.874i)11-s + (0.269 + 0.467i)13-s + 0.563·17-s − 1.55·19-s + (0.933 + 1.61i)23-s + (0.401 − 0.694i)25-s + (0.452 − 0.784i)29-s + (−0.555 − 0.962i)31-s − 0.241·35-s + 0.0272·37-s + (0.842 + 1.45i)41-s + (−0.679 + 1.17i)43-s + (−0.477 + 0.826i)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: $\chi_{2916} (1945, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2916,\ (\ :1/2),\ 0.5 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.411423185\)
\(L(\frac12)\) \(\approx\) \(1.411423185\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.497 + 0.861i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.719 + 1.24i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.67 - 2.89i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.972 - 1.68i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.32T + 17T^{2} \)
19 \( 1 + 6.77T + 19T^{2} \)
23 \( 1 + (-4.47 - 7.75i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.43 + 4.22i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.09 + 5.36i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.165T + 37T^{2} \)
41 \( 1 + (-5.39 - 9.34i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.45 - 7.72i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.27 - 5.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 + (-1.97 - 3.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.21 - 7.29i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0110 - 0.0191i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.07T + 71T^{2} \)
73 \( 1 - 4.53T + 73T^{2} \)
79 \( 1 + (-2.89 + 5.01i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.58 + 4.48i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.83T + 89T^{2} \)
97 \( 1 + (-3.53 + 6.11i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.915168981146008355992567635065, −7.86573384807777690427026450031, −7.68563850913235374471582233572, −6.58825537131130161491313750669, −5.90837693013521976457741986194, −4.63310996644454494662978219982, −4.50520507612872220459868569798, −3.30752812128406700714501415459, −2.16831945818325580123820077752, −1.11032423137268259676961672680, 0.49911024912817588582850663668, 2.00922870027322957710427922092, 3.00546250061631864603271350521, 3.68160565457360279698689838973, 4.93283038895842875761329423671, 5.44957900612899094112909795790, 6.47088567173553796695164114233, 6.99860474402054809188844454519, 8.112123334259305939679410494091, 8.545946485965440609973500124787

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