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Properties

Label 1-4136-4136.3101-r0-0-0
Degree $1$
Conductor $4136$
Sign $1$
Analytic cond. $19.2075$
Root an. cond. $19.2075$
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  − 3-s + 5-s − 7-s + 9-s − 13-s − 15-s − 17-s − 19-s + 21-s − 23-s + 25-s − 27-s − 29-s − 31-s − 35-s − 37-s + 39-s + 41-s − 43-s + 45-s + 49-s + 51-s − 53-s + 57-s − 59-s + 61-s − 63-s + ⋯
L(s)  = 1  − 3-s + 5-s − 7-s + 9-s − 13-s − 15-s − 17-s − 19-s + 21-s − 23-s + 25-s − 27-s − 29-s − 31-s − 35-s − 37-s + 39-s + 41-s − 43-s + 45-s + 49-s + 51-s − 53-s + 57-s − 59-s + 61-s − 63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4136 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4136 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4136\)    =    \(2^{3} \cdot 11 \cdot 47\)
Sign: $1$
Analytic conductor: \(19.2075\)
Root analytic conductor: \(19.2075\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4136} (3101, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 4136,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6184494867\)
\(L(\frac12)\) \(\approx\) \(0.6184494867\)
\(L(1)\) \(\approx\) \(0.6447518044\)
\(L(1)\) \(\approx\) \(0.6447518044\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
47 \( 1 \)
good3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 + T \)
43 \( 1 - T \)
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

−18.39006669778179866847983890872, −17.49540443194634878823756971248, −17.19451698916280937979377134257, −16.51846879670207946309304407038, −15.882988787416239486819441808686, −15.12722101427672905636894660331, −14.31139661039885060084868006101, −13.45929577945449238136515685002, −12.75653141664725640777140912727, −12.553808779304883983774681759222, −11.536591869531205823906759621811, −10.72516094958905559407485662734, −10.23133802297319960222727002917, −9.50225493011533538266973692531, −9.074505083658721668501831290714, −7.83305983324762585413984338305, −6.839693577417933288969847041554, −6.54014729044746880335148887793, −5.78017954516808491561043769352, −5.15361097367966947081598690584, −4.3320261316488419851686875782, −3.47477073449256949542734334529, −2.2336689886377190105451027780, −1.86927538173006512164899366897, −0.41650860202469570622957159176, 0.41650860202469570622957159176, 1.86927538173006512164899366897, 2.2336689886377190105451027780, 3.47477073449256949542734334529, 4.3320261316488419851686875782, 5.15361097367966947081598690584, 5.78017954516808491561043769352, 6.54014729044746880335148887793, 6.839693577417933288969847041554, 7.83305983324762585413984338305, 9.074505083658721668501831290714, 9.50225493011533538266973692531, 10.23133802297319960222727002917, 10.72516094958905559407485662734, 11.536591869531205823906759621811, 12.553808779304883983774681759222, 12.75653141664725640777140912727, 13.45929577945449238136515685002, 14.31139661039885060084868006101, 15.12722101427672905636894660331, 15.882988787416239486819441808686, 16.51846879670207946309304407038, 17.19451698916280937979377134257, 17.49540443194634878823756971248, 18.39006669778179866847983890872

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