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Properties

Label 1-197-197.23-r0-0-0
Degree $1$
Conductor $197$
Sign $0.156 + 0.987i$
Analytic cond. $0.914864$
Root an. cond. $0.914864$
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  + (−0.761 − 0.648i)2-s + (0.404 − 0.914i)3-s + (0.159 + 0.987i)4-s + (−0.997 + 0.0640i)5-s + (−0.900 + 0.433i)6-s + (−0.761 + 0.648i)7-s + (0.518 − 0.855i)8-s + (−0.672 − 0.740i)9-s + (0.801 + 0.598i)10-s + (0.0320 − 0.999i)11-s + (0.967 + 0.253i)12-s + (−0.345 + 0.938i)13-s + 14-s + (−0.345 + 0.938i)15-s + (−0.949 + 0.315i)16-s + (−0.572 + 0.820i)17-s + ⋯
L(s)  = 1  + (−0.761 − 0.648i)2-s + (0.404 − 0.914i)3-s + (0.159 + 0.987i)4-s + (−0.997 + 0.0640i)5-s + (−0.900 + 0.433i)6-s + (−0.761 + 0.648i)7-s + (0.518 − 0.855i)8-s + (−0.672 − 0.740i)9-s + (0.801 + 0.598i)10-s + (0.0320 − 0.999i)11-s + (0.967 + 0.253i)12-s + (−0.345 + 0.938i)13-s + 14-s + (−0.345 + 0.938i)15-s + (−0.949 + 0.315i)16-s + (−0.572 + 0.820i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(197\)
Sign: $0.156 + 0.987i$
Analytic conductor: \(0.914864\)
Root analytic conductor: \(0.914864\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{197} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 197,\ (0:\ ),\ 0.156 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1340291161 + 0.1145068805i\)
\(L(\frac12)\) \(\approx\) \(0.1340291161 + 0.1145068805i\)
\(L(1)\) \(\approx\) \(0.4561842432 - 0.1592519473i\)
\(L(1)\) \(\approx\) \(0.4561842432 - 0.1592519473i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad197 \( 1 \)
good2 \( 1 + (-0.761 - 0.648i)T \)
3 \( 1 + (0.404 - 0.914i)T \)
5 \( 1 + (-0.997 + 0.0640i)T \)
7 \( 1 + (-0.761 + 0.648i)T \)
11 \( 1 + (0.0320 - 0.999i)T \)
13 \( 1 + (-0.345 + 0.938i)T \)
17 \( 1 + (-0.572 + 0.820i)T \)
19 \( 1 + (-0.222 + 0.974i)T \)
23 \( 1 + (-0.981 + 0.191i)T \)
29 \( 1 + (0.967 + 0.253i)T \)
31 \( 1 + (-0.462 + 0.886i)T \)
37 \( 1 + (-0.949 - 0.315i)T \)
41 \( 1 + (-0.572 + 0.820i)T \)
43 \( 1 + (0.0320 - 0.999i)T \)
47 \( 1 + (-0.838 + 0.545i)T \)
53 \( 1 + (0.871 + 0.490i)T \)
59 \( 1 + (0.801 - 0.598i)T \)
61 \( 1 + (0.404 + 0.914i)T \)
67 \( 1 + (-0.838 + 0.545i)T \)
71 \( 1 + (-0.672 - 0.740i)T \)
73 \( 1 + (-0.949 - 0.315i)T \)
79 \( 1 + (-0.997 - 0.0640i)T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (-0.462 - 0.886i)T \)
97 \( 1 + (0.284 + 0.958i)T \)
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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

−26.62553100769254998037255529767, −26.0735215090624454006347928815, −25.159212231467215412655101747862, −24.02145789847325374185173556841, −22.859332133685983168410891791845, −22.44334845317907806119323469360, −20.481670797655638818740198974659, −19.96895234663833449262343788871, −19.38872976614651819817670111926, −17.919208191360897412723407501252, −16.92517062076387692995892321235, −15.88200631508394760555195713068, −15.4938481884215980645489691125, −14.532780331577979508764604743233, −13.256893793848729953229215353650, −11.6478226166608535918300855332, −10.45180737575328067227310673114, −9.803869315704314586813325722917, −8.70339843172602080220276678838, −7.6692187394983704065632905997, −6.79660622790087006450043279591, −5.07575390207796989001797219748, −4.139840712486926824079513179068, −2.64970646822444866865036957904, −0.15955935588251719122457857558, 1.69723413644946096253304230266, 3.00196685389248242913820956379, 3.883781187628197713024030811905, 6.2316168466596282062900616227, 7.190612715092041589895623037390, 8.442253379418624404246320720203, 8.80532934817924822112851114284, 10.33151539183324388380017498796, 11.69952782140162811118826385980, 12.16030757879445042463374031159, 13.14471351665028100458083649771, 14.41086615358780172816385252857, 15.83165159157726814922833289075, 16.61362516788648633541815494482, 17.958589803942375246643856297, 18.95016401867686603582037880573, 19.29195895676098883601769536289, 20.00697944391261861459867181189, 21.356792879920602870306813019661, 22.28218153219039702482245606048, 23.54785663107414044106860058894, 24.44046889901231106631049130959, 25.46911902780751582916449493118, 26.35269681978162471378637778694, 27.05527312768358012137705641720

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