Properties

Label 1-18e2-324.47-r0-0-0
Degree $1$
Conductor $324$
Sign $0.999 + 0.0193i$
Analytic cond. $1.50464$
Root an. cond. $1.50464$
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  + (0.993 − 0.116i)5-s + (−0.893 − 0.448i)7-s + (0.396 + 0.918i)11-s + (0.973 + 0.230i)13-s + (−0.173 + 0.984i)17-s + (−0.173 − 0.984i)19-s + (0.893 − 0.448i)23-s + (0.973 − 0.230i)25-s + (0.286 − 0.957i)29-s + (0.835 + 0.549i)31-s + (−0.939 − 0.342i)35-s + (−0.939 + 0.342i)37-s + (0.686 − 0.727i)41-s + (−0.597 + 0.802i)43-s + (−0.835 + 0.549i)47-s + ⋯
L(s)  = 1  + (0.993 − 0.116i)5-s + (−0.893 − 0.448i)7-s + (0.396 + 0.918i)11-s + (0.973 + 0.230i)13-s + (−0.173 + 0.984i)17-s + (−0.173 − 0.984i)19-s + (0.893 − 0.448i)23-s + (0.973 − 0.230i)25-s + (0.286 − 0.957i)29-s + (0.835 + 0.549i)31-s + (−0.939 − 0.342i)35-s + (−0.939 + 0.342i)37-s + (0.686 − 0.727i)41-s + (−0.597 + 0.802i)43-s + (−0.835 + 0.549i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.999 + 0.0193i$
Analytic conductor: \(1.50464\)
Root analytic conductor: \(1.50464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 324,\ (0:\ ),\ 0.999 + 0.0193i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.459562774 + 0.01415276356i\)
\(L(\frac12)\) \(\approx\) \(1.459562774 + 0.01415276356i\)
\(L(1)\) \(\approx\) \(1.217044015 + 0.01358326583i\)
\(L(1)\) \(\approx\) \(1.217044015 + 0.01358326583i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.993 - 0.116i)T \)
7 \( 1 + (-0.893 - 0.448i)T \)
11 \( 1 + (0.396 + 0.918i)T \)
13 \( 1 + (0.973 + 0.230i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (0.893 - 0.448i)T \)
29 \( 1 + (0.286 - 0.957i)T \)
31 \( 1 + (0.835 + 0.549i)T \)
37 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (0.686 - 0.727i)T \)
43 \( 1 + (-0.597 + 0.802i)T \)
47 \( 1 + (-0.835 + 0.549i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.396 - 0.918i)T \)
61 \( 1 + (-0.0581 - 0.998i)T \)
67 \( 1 + (0.286 + 0.957i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (0.686 + 0.727i)T \)
83 \( 1 + (-0.686 - 0.727i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
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

−25.15846503511425238527606519364, −24.500859516523834437689515806368, −23.060101883934833628590790051520, −22.497572270731073211662299864309, −21.476800655961435070201670858737, −20.88364990254649407120866766650, −19.662031400460767627372280938627, −18.681791297333425710225904786066, −18.11476840586071854170155634531, −16.84346078599977596864684055220, −16.22077877368883712258411276624, −15.15289099644355600285593907681, −13.91098130213563010836809760432, −13.40925357599554081460698759180, −12.35482159484876062624400440155, −11.18978916517479199495991981845, −10.196734055453617745882248109036, −9.24072599226454999579613304598, −8.524720087403353165687217213805, −6.89363839026545338189567477582, −6.08467267261366768519661340006, −5.31265561697799691187237161904, −3.585251927441965739101983820405, −2.714486319172802342513878452449, −1.223081869829802086079008451600, 1.255918522491785024847743970988, 2.52702334092917782070118054093, 3.85035342513974717656371728409, 4.98916600109110329858231108298, 6.43661370657878363171663861433, 6.72071025809485787023948522622, 8.419716924296204596927779997848, 9.38466344040033595453615199485, 10.148174664877282061659680446854, 11.09080259076045087572390205367, 12.54980842121914842437882823944, 13.185413751991515378458194725198, 13.999185410598119111330275122162, 15.14170579888801449509649040688, 16.10745804994672267080489504545, 17.2139593334475190343083047273, 17.61966762360446882995412655614, 18.91740521551607724797122661692, 19.7216344379284034538704022726, 20.742851367263791131861384580674, 21.46299756173556276134317757493, 22.54335099181887979353025833104, 23.13681349539910167599507619826, 24.29552278891018227898861673259, 25.26447986215861752909187881769

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