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Properties

Label 1-197-197.178-r0-0-0
Degree $1$
Conductor $197$
Sign $-0.466 + 0.884i$
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  + (−0.222 + 0.974i)2-s + (−0.222 − 0.974i)3-s + (−0.900 − 0.433i)4-s + (−0.900 + 0.433i)5-s + 6-s + (−0.222 − 0.974i)7-s + (0.623 − 0.781i)8-s + (−0.900 + 0.433i)9-s + (−0.222 − 0.974i)10-s + (−0.222 + 0.974i)11-s + (−0.222 + 0.974i)12-s + (0.623 + 0.781i)13-s + 14-s + (0.623 + 0.781i)15-s + (0.623 + 0.781i)16-s + (−0.900 + 0.433i)17-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−0.222 − 0.974i)3-s + (−0.900 − 0.433i)4-s + (−0.900 + 0.433i)5-s + 6-s + (−0.222 − 0.974i)7-s + (0.623 − 0.781i)8-s + (−0.900 + 0.433i)9-s + (−0.222 − 0.974i)10-s + (−0.222 + 0.974i)11-s + (−0.222 + 0.974i)12-s + (0.623 + 0.781i)13-s + 14-s + (0.623 + 0.781i)15-s + (0.623 + 0.781i)16-s + (−0.900 + 0.433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(197\)
Sign: $-0.466 + 0.884i$
Analytic conductor: \(0.914864\)
Root analytic conductor: \(0.914864\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{197} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 197,\ (0:\ ),\ -0.466 + 0.884i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2384978354 + 0.3954372304i\)
\(L(\frac12)\) \(\approx\) \(0.2384978354 + 0.3954372304i\)
\(L(1)\) \(\approx\) \(0.5593111513 + 0.2082259236i\)
\(L(1)\) \(\approx\) \(0.5593111513 + 0.2082259236i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad197 \( 1 \)
good2 \( 1 + (-0.222 + 0.974i)T \)
3 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + (-0.222 - 0.974i)T \)
11 \( 1 + (-0.222 + 0.974i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + (-0.900 + 0.433i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (-0.222 + 0.974i)T \)
31 \( 1 + (-0.222 + 0.974i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (0.623 - 0.781i)T \)
53 \( 1 + (-0.900 - 0.433i)T \)
59 \( 1 + (-0.222 + 0.974i)T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 + (0.623 - 0.781i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (0.623 - 0.781i)T \)
79 \( 1 + (-0.900 - 0.433i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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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

−26.96644931772141012822011055511, −26.20204552971882096099314318390, −24.78711873549386623346783218307, −23.513030353740799976977090197199, −22.32470467967205792567995631160, −22.090961803494697863981374231645, −20.592712352684715158547520836215, −20.39524156518326344967849291333, −19.03632455580091309651956582071, −18.283328036055316875379839403813, −16.96381441367404361924502484388, −15.94196777330811458365279280484, −15.30353221509038269523637113064, −13.77087457894680754619718020881, −12.5900538260024035319527965311, −11.54699764097185710301381450255, −11.05644309204430721087298136471, −9.74227050567653396274598899694, −8.769979701286052513262162135685, −8.12877411981355181025032830674, −5.82800168295790536836008263642, −4.79570517811356155393971110945, −3.64272659156992358887487790000, −2.75289425753691861451761808956, −0.43061588263168657741121720337, 1.419211748561737514700917707379, 3.60307756983912166054286175259, 4.82333638099081997897933401531, 6.39113056022853772164235513613, 7.16785524766510915370500226169, 7.70384368503067030872476166661, 8.970335442862803752868415782084, 10.43839098384210502935535452772, 11.511535015843865760172703390, 12.82620209464029318919925129373, 13.71484108285143635065477612726, 14.618298693827788355632309842293, 15.80896758562483555182269576627, 16.63136825514333388638932158393, 17.808292396938567113217575236332, 18.32879538108787809990168617755, 19.55760284614066869913261299952, 20.02810058330456007614173267162, 22.10428183977973141912415885688, 23.11809623750161505750203258080, 23.4933118663253657191761734801, 24.2187564576422409361176599616, 25.47572188658802977188062537056, 26.189991339785700843143131945833, 26.98920940433235998708314740781

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