Properties

Label 1-888-888.443-r0-0-0
Degree $1$
Conductor $888$
Sign $1$
Analytic cond. $4.12385$
Root an. cond. $4.12385$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 7-s − 11-s + 13-s + 17-s − 19-s − 23-s + 25-s − 29-s + 31-s + 35-s − 41-s − 43-s + 47-s + 49-s + 53-s + 55-s + 59-s + 61-s − 65-s + 67-s + 71-s + 73-s + 77-s + 79-s − 83-s − 85-s + ⋯
L(s)  = 1  − 5-s − 7-s − 11-s + 13-s + 17-s − 19-s − 23-s + 25-s − 29-s + 31-s + 35-s − 41-s − 43-s + 47-s + 49-s + 53-s + 55-s + 59-s + 61-s − 65-s + 67-s + 71-s + 73-s + 77-s + 79-s − 83-s − 85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $1$
Analytic conductor: \(4.12385\)
Root analytic conductor: \(4.12385\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{888} (443, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 888,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8339061315\)
\(L(\frac12)\) \(\approx\) \(0.8339061315\)
\(L(1)\) \(\approx\) \(0.7647262465\)
\(L(1)\) \(\approx\) \(0.7647262465\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
37 \( 1 \)
good5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + T \)
97 \( 1 - T \)
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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

−22.07307341717427806096322533583, −21.06863621609457198871778882804, −20.38526734593996537875948077322, −19.542357141289456915399824188669, −18.74697446833285997413832518801, −18.39428928871324612229074984933, −16.97600455798195755433971868978, −16.28894323189071509193665134451, −15.63117826661095320236017687597, −15.037327719661911296272572132134, −13.819524871108239076177305984079, −13.02233859497537813940221226953, −12.34055463930161833290543613736, −11.47046595860794759574162752200, −10.511481255161369998780437466245, −9.88318892599323320858129528317, −8.59304642647766950390646343130, −8.06189314239177612439331938362, −7.063323417266076539608008029881, −6.17371621105540805922921636073, −5.22533926257239501912600128391, −3.93678107772296455176712461049, −3.43432821460292707745767010929, −2.288873980769206398450116464015, −0.652142048486067845636538846296, 0.652142048486067845636538846296, 2.288873980769206398450116464015, 3.43432821460292707745767010929, 3.93678107772296455176712461049, 5.22533926257239501912600128391, 6.17371621105540805922921636073, 7.063323417266076539608008029881, 8.06189314239177612439331938362, 8.59304642647766950390646343130, 9.88318892599323320858129528317, 10.511481255161369998780437466245, 11.47046595860794759574162752200, 12.34055463930161833290543613736, 13.02233859497537813940221226953, 13.819524871108239076177305984079, 15.037327719661911296272572132134, 15.63117826661095320236017687597, 16.28894323189071509193665134451, 16.97600455798195755433971868978, 18.39428928871324612229074984933, 18.74697446833285997413832518801, 19.542357141289456915399824188669, 20.38526734593996537875948077322, 21.06863621609457198871778882804, 22.07307341717427806096322533583

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