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Properties

Label 1-72e2-5184.371-r0-0-0
Degree $1$
Conductor $5184$
Sign $0.706 - 0.707i$
Analytic cond. $24.0743$
Root an. cond. $24.0743$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.873 − 0.487i)5-s + (0.949 − 0.314i)7-s + (0.696 − 0.717i)11-s + (0.810 + 0.585i)13-s + (−0.984 − 0.173i)17-s + (−0.216 + 0.976i)19-s + (0.949 + 0.314i)23-s + (0.524 − 0.851i)25-s + (0.775 + 0.631i)29-s + (−0.835 − 0.549i)31-s + (0.675 − 0.737i)35-s + (0.737 − 0.675i)37-s + (0.999 − 0.0290i)41-s + (−0.969 − 0.244i)43-s + (−0.549 − 0.835i)47-s + ⋯
L(s)  = 1  + (0.873 − 0.487i)5-s + (0.949 − 0.314i)7-s + (0.696 − 0.717i)11-s + (0.810 + 0.585i)13-s + (−0.984 − 0.173i)17-s + (−0.216 + 0.976i)19-s + (0.949 + 0.314i)23-s + (0.524 − 0.851i)25-s + (0.775 + 0.631i)29-s + (−0.835 − 0.549i)31-s + (0.675 − 0.737i)35-s + (0.737 − 0.675i)37-s + (0.999 − 0.0290i)41-s + (−0.969 − 0.244i)43-s + (−0.549 − 0.835i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $0.706 - 0.707i$
Analytic conductor: \(24.0743\)
Root analytic conductor: \(24.0743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5184,\ (0:\ ),\ 0.706 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.622467559 - 1.088124031i\)
\(L(\frac12)\) \(\approx\) \(2.622467559 - 1.088124031i\)
\(L(1)\) \(\approx\) \(1.502245008 - 0.2589842753i\)
\(L(1)\) \(\approx\) \(1.502245008 - 0.2589842753i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.873 - 0.487i)T \)
7 \( 1 + (0.949 - 0.314i)T \)
11 \( 1 + (0.696 - 0.717i)T \)
13 \( 1 + (0.810 + 0.585i)T \)
17 \( 1 + (-0.984 - 0.173i)T \)
19 \( 1 + (-0.216 + 0.976i)T \)
23 \( 1 + (0.949 + 0.314i)T \)
29 \( 1 + (0.775 + 0.631i)T \)
31 \( 1 + (-0.835 - 0.549i)T \)
37 \( 1 + (0.737 - 0.675i)T \)
41 \( 1 + (0.999 - 0.0290i)T \)
43 \( 1 + (-0.969 - 0.244i)T \)
47 \( 1 + (-0.549 - 0.835i)T \)
53 \( 1 + (-0.991 - 0.130i)T \)
59 \( 1 + (0.0145 - 0.999i)T \)
61 \( 1 + (-0.944 - 0.328i)T \)
67 \( 1 + (0.994 + 0.101i)T \)
71 \( 1 + (-0.996 + 0.0871i)T \)
73 \( 1 + (-0.996 - 0.0871i)T \)
79 \( 1 + (0.727 - 0.686i)T \)
83 \( 1 + (0.355 + 0.934i)T \)
89 \( 1 + (-0.0871 + 0.996i)T \)
97 \( 1 + (0.993 + 0.116i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.05066492998158763847916395607, −17.42502905083010705533777236716, −17.111621344281597373200570648828, −15.959912399844072759827689899869, −15.2433480976275742969611925966, −14.75924260608103171780725899127, −14.21086685952162991161507085446, −13.2844841432233706251907737407, −12.99567320702879565869000296566, −11.97372806648383688533781935999, −11.136690091189120045376009647228, −10.87707810597538545806971267785, −10.02198673762250506438722261011, −9.12193871651264217161211678483, −8.80342963044657819231352550226, −7.87067357347416980507294408219, −7.0227011426370328875724076819, −6.400031867996611088725693476854, −5.807270354574506618395479011903, −4.758769446990928207867557558201, −4.48202716735941902008609094000, −3.19687432775314167216099974326, −2.53375064874880834745503494063, −1.734445466283724716777365866426, −1.07905290086342738110348249743, 0.806723024505113303048050433122, 1.57967111677149073827898153620, 2.06103271282202958582205573624, 3.27336342428396656969260212856, 4.08383408835857402648615096166, 4.74936965089411755666348949530, 5.50257886014311739818843840147, 6.2451940761327762391499696858, 6.78956902687636280428374742410, 7.81228000908080114523240499212, 8.5979313991804638195789429815, 8.987440663961725726131279643532, 9.68399293387407270734025975747, 10.708041299711391589852363310257, 11.117462668881941853610286820066, 11.75411692891010040157687744952, 12.71283143630286644251495089878, 13.31721362034541542999777823622, 13.98785701995791975736651098776, 14.38040955809549214772607977990, 15.12360280910491604200574186404, 16.23037564302433936636236543677, 16.544448451038063422451684032050, 17.25542535009591694048574647349, 17.8556212008182825681428540774

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