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Properties

Label 1-1441-1441.137-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.573 + 0.819i$
Analytic cond. $154.856$
Root an. cond. $154.856$
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  + (0.970 − 0.239i)2-s + (−0.681 − 0.732i)3-s + (0.885 − 0.464i)4-s + (−0.906 + 0.421i)5-s + (−0.836 − 0.548i)6-s + (−0.262 − 0.964i)7-s + (0.748 − 0.663i)8-s + (−0.0724 + 0.997i)9-s + (−0.779 + 0.626i)10-s + (−0.943 − 0.331i)12-s + (−0.998 − 0.0483i)13-s + (−0.485 − 0.873i)14-s + (0.926 + 0.377i)15-s + (0.568 − 0.822i)16-s + (−0.0241 − 0.999i)17-s + (0.168 + 0.985i)18-s + ⋯
L(s)  = 1  + (0.970 − 0.239i)2-s + (−0.681 − 0.732i)3-s + (0.885 − 0.464i)4-s + (−0.906 + 0.421i)5-s + (−0.836 − 0.548i)6-s + (−0.262 − 0.964i)7-s + (0.748 − 0.663i)8-s + (−0.0724 + 0.997i)9-s + (−0.779 + 0.626i)10-s + (−0.943 − 0.331i)12-s + (−0.998 − 0.0483i)13-s + (−0.485 − 0.873i)14-s + (0.926 + 0.377i)15-s + (0.568 − 0.822i)16-s + (−0.0241 − 0.999i)17-s + (0.168 + 0.985i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.573 + 0.819i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.573 + 0.819i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6489558268 - 1.246829084i\)
\(L(\frac12)\) \(\approx\) \(-0.6489558268 - 1.246829084i\)
\(L(1)\) \(\approx\) \(0.9218864425 - 0.7279938704i\)
\(L(1)\) \(\approx\) \(0.9218864425 - 0.7279938704i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.970 - 0.239i)T \)
3 \( 1 + (-0.681 - 0.732i)T \)
5 \( 1 + (-0.906 + 0.421i)T \)
7 \( 1 + (-0.262 - 0.964i)T \)
13 \( 1 + (-0.998 - 0.0483i)T \)
17 \( 1 + (-0.0241 - 0.999i)T \)
19 \( 1 + (0.607 - 0.794i)T \)
23 \( 1 + (0.861 - 0.506i)T \)
29 \( 1 + (-0.926 + 0.377i)T \)
31 \( 1 + (-0.399 - 0.916i)T \)
37 \( 1 + (0.443 - 0.896i)T \)
41 \( 1 + (0.958 - 0.285i)T \)
43 \( 1 + (-0.681 - 0.732i)T \)
47 \( 1 + (0.527 + 0.849i)T \)
53 \( 1 + T \)
59 \( 1 + (0.568 - 0.822i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (0.681 - 0.732i)T \)
71 \( 1 + (-0.995 + 0.0965i)T \)
73 \( 1 - T \)
79 \( 1 + (0.681 + 0.732i)T \)
83 \( 1 + (0.168 - 0.985i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.779 - 0.626i)T \)
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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

−21.3423256651357245209352331821, −20.46866130859127859315779603430, −19.7227350817367956590891181476, −18.947310296695601975815283861246, −17.791888087050425963894953145215, −16.78324591806571428073796294367, −16.46777198918216435508567434660, −15.612113531258337190017634864716, −14.966454512861890950645482628915, −14.69904609171529337801870095529, −13.16490686823908004876043381960, −12.49213965230710560774963361985, −11.91132479183446774224009928660, −11.41561725349945955005998590907, −10.45140420217184932478823396023, −9.46067306840997519305534349661, −8.54223582853981567581872941937, −7.60377571491953686763475040538, −6.7135463035390701477855877112, −5.686665275097402420379756953623, −5.246394919632231285503258868018, −4.37649186068726338601058316545, −3.61157964721865140842233865134, −2.82933815312068475058172591428, −1.4092528010462104687764754165, 0.24948441992234426371558623850, 0.880394456019193679777470030659, 2.32262913666287262019285242296, 3.05718898188217034791302180366, 4.15345467116630656966185763376, 4.8399654283836000500647304608, 5.69759592790055787227804365015, 6.90652002644933674525572707603, 7.19306113883141212424129247135, 7.70564185842688756475187662320, 9.38642441765612351317266630388, 10.46553833290953911472630240826, 11.11707254182546363588666342901, 11.597474578227643619677502185882, 12.46386977676386588694941637580, 13.05259430463869028996891769145, 13.86522190118402855640733303235, 14.560662002086323048229115754703, 15.42503358793403254774520609278, 16.35200372794475669816357305127, 16.762286599271581483734407645984, 17.84568086684950756665361725517, 18.7865426730396604658442904701, 19.39914552812649431532910652423, 20.0395203611886868133716057232

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