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Properties

Label 1-48e2-2304.173-r1-0-0
Degree $1$
Conductor $2304$
Sign $0.709 + 0.705i$
Analytic cond. $247.599$
Root an. cond. $247.599$
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  + (−0.935 + 0.352i)5-s + (0.0654 − 0.997i)7-s + (0.973 − 0.227i)11-s + (0.986 − 0.162i)13-s + (−0.923 − 0.382i)17-s + (0.995 + 0.0980i)19-s + (−0.659 + 0.751i)23-s + (0.751 − 0.659i)25-s + (0.729 + 0.683i)29-s + (0.258 + 0.965i)31-s + (0.290 + 0.956i)35-s + (0.0980 + 0.995i)37-s + (−0.751 − 0.659i)41-s + (0.528 − 0.849i)43-s + (0.991 − 0.130i)47-s + ⋯
L(s)  = 1  + (−0.935 + 0.352i)5-s + (0.0654 − 0.997i)7-s + (0.973 − 0.227i)11-s + (0.986 − 0.162i)13-s + (−0.923 − 0.382i)17-s + (0.995 + 0.0980i)19-s + (−0.659 + 0.751i)23-s + (0.751 − 0.659i)25-s + (0.729 + 0.683i)29-s + (0.258 + 0.965i)31-s + (0.290 + 0.956i)35-s + (0.0980 + 0.995i)37-s + (−0.751 − 0.659i)41-s + (0.528 − 0.849i)43-s + (0.991 − 0.130i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.709 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.709 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.709 + 0.705i$
Analytic conductor: \(247.599\)
Root analytic conductor: \(247.599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2304,\ (1:\ ),\ 0.709 + 0.705i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.663235289 + 0.6862791195i\)
\(L(\frac12)\) \(\approx\) \(1.663235289 + 0.6862791195i\)
\(L(1)\) \(\approx\) \(1.011656262 + 0.007060002092i\)
\(L(1)\) \(\approx\) \(1.011656262 + 0.007060002092i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.935 + 0.352i)T \)
7 \( 1 + (0.0654 - 0.997i)T \)
11 \( 1 + (0.973 - 0.227i)T \)
13 \( 1 + (0.986 - 0.162i)T \)
17 \( 1 + (-0.923 - 0.382i)T \)
19 \( 1 + (0.995 + 0.0980i)T \)
23 \( 1 + (-0.659 + 0.751i)T \)
29 \( 1 + (0.729 + 0.683i)T \)
31 \( 1 + (0.258 + 0.965i)T \)
37 \( 1 + (0.0980 + 0.995i)T \)
41 \( 1 + (-0.751 - 0.659i)T \)
43 \( 1 + (0.528 - 0.849i)T \)
47 \( 1 + (0.991 - 0.130i)T \)
53 \( 1 + (-0.956 - 0.290i)T \)
59 \( 1 + (0.352 + 0.935i)T \)
61 \( 1 + (0.0327 + 0.999i)T \)
67 \( 1 + (-0.849 + 0.528i)T \)
71 \( 1 + (-0.555 + 0.831i)T \)
73 \( 1 + (0.831 - 0.555i)T \)
79 \( 1 + (-0.608 + 0.793i)T \)
83 \( 1 + (0.910 - 0.412i)T \)
89 \( 1 + (0.980 - 0.195i)T \)
97 \( 1 + (0.965 + 0.258i)T \)
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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

−19.33313830208236555224979734084, −18.735278161688495949336993024886, −17.99236720351021769033267933126, −17.26531319912139745976570477743, −16.24613809145438904562341633017, −15.82169149593444253051778965687, −15.17945175600655417347928450129, −14.42864750238140200637362387948, −13.565772084922851981634462931078, −12.67115546469477101029185500858, −12.016805321132136245301840272542, −11.48730859795458114157344989846, −10.83993365840925257991034518531, −9.55875191259376353228642569154, −9.01950799297304467366051340251, −8.30778130243694395261260534565, −7.67989074780120914886951211900, −6.47723836742237457936279877922, −6.08286570884701740606893107349, −4.87863936394146031830461831883, −4.215199534992196751284013037552, −3.472041226314657535854520606731, −2.41321695299216346011110232929, −1.43666061392968219427589225654, −0.415263980831504544922619848553, 0.791214327140586387310002322392, 1.433771901486704036303517165417, 2.93968148303394136063358183223, 3.677479911812340608176819806326, 4.16774489437769129906868868686, 5.13887842565911843510573661941, 6.32282084111378023617578812129, 6.947207499554705595509903589750, 7.56863532297361729197883185653, 8.482048041958705905846609636452, 9.08771034601432882680898480051, 10.26978768929259781961592882147, 10.74555297412742113544391670344, 11.69821980042202533688016030891, 11.91353985609966898919538982329, 13.19476393209675565792595277331, 13.89287070102817613017070153560, 14.30209127515067775313699001259, 15.42807130074720318481288672114, 15.91503072414831420525068941164, 16.53762477310572987082106526282, 17.50712475016148361143028246791, 18.04365751102061969089884130888, 18.92707815332801304285628031068, 19.66575447103132896423875170488

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