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Properties

Label 1-31-31.29-r1-0-0
Degree $1$
Conductor $31$
Sign $-0.943 + 0.330i$
Analytic cond. $3.33141$
Root an. cond. $3.33141$
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  + (0.309 + 0.951i)2-s + (−0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s + 5-s − 6-s + (−0.809 + 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.309 + 0.951i)10-s + (0.809 − 0.587i)11-s + (−0.309 − 0.951i)12-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)14-s + (−0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s + 5-s − 6-s + (−0.809 + 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.309 + 0.951i)10-s + (0.809 − 0.587i)11-s + (−0.309 − 0.951i)12-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)14-s + (−0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(31\)
Sign: $-0.943 + 0.330i$
Analytic conductor: \(3.33141\)
Root analytic conductor: \(3.33141\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 31,\ (1:\ ),\ -0.943 + 0.330i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2306052147 + 1.355289584i\)
\(L(\frac12)\) \(\approx\) \(0.2306052147 + 1.355289584i\)
\(L(1)\) \(\approx\) \(0.6840578398 + 0.8975311770i\)
\(L(1)\) \(\approx\) \(0.6840578398 + 0.8975311770i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 - T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−36.3264916760952652019335036877, −35.06305480502410161681708577102, −33.20776651101042650124611929390, −32.24583830347167259813804629142, −30.55530566514374416496190176640, −29.70610997447390697643991575847, −29.01989183387248625639754731468, −27.737335621005811522296759924430, −25.810042929625683137730259843335, −24.54461399147380580276791697674, −22.92529958625581467838012734333, −22.281538155632123349105450736603, −20.4802632716753794249522081216, −19.43194999484478169548046725163, −18.05847051770838454567063996815, −17.05730152329900190669343375002, −14.3579034694328417477601062636, −13.237465302381965744828803824344, −12.347433074064011641032895259365, −10.62398288545876519417484094782, −9.303467786498165975817208841000, −6.87084990654634285017173513537, −5.31067982660713559736577140637, −2.83890296732804135213506855452, −1.01173093662613643426242176671, 3.57398868718491103235965839590, 5.46047509861528011814364919438, 6.40162584038573998066340663304, 8.93914514224797413478561429479, 9.82577459543882685780199649961, 12.09187218286082418930680669002, 13.8318583593302528089913574979, 14.97681799438143507069103273896, 16.47664793802865751004477724308, 17.09994647754203425381366651929, 18.85275866850004954839123929005, 21.2525173222356469469254671812, 21.91969548709758263463953940883, 23.00984372852617785178832465789, 24.75535993199169037760916238648, 25.771046079438126957881227459803, 26.81378634133369164224485359562, 28.232037283561745608367441575668, 29.53993736867000404763415161624, 31.53576512828425439738789013605, 32.4440937612587330538639950522, 33.29665436203929818022593258667, 34.33921375711091180796854132469, 35.49714373668704611137346663830, 37.066473861800698913391030712399

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