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Properties

Label 1-197-197.34-r0-0-0
Degree $1$
Conductor $197$
Sign $-0.806 + 0.591i$
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  + (0.404 − 0.914i)2-s + (0.0320 − 0.999i)3-s + (−0.672 − 0.740i)4-s + (−0.572 − 0.820i)5-s + (−0.900 − 0.433i)6-s + (0.404 + 0.914i)7-s + (−0.949 + 0.315i)8-s + (−0.997 − 0.0640i)9-s + (−0.981 + 0.191i)10-s + (−0.462 − 0.886i)11-s + (−0.761 + 0.648i)12-s + (−0.838 + 0.545i)13-s + 14-s + (−0.838 + 0.545i)15-s + (−0.0960 + 0.995i)16-s + (0.284 − 0.958i)17-s + ⋯
L(s)  = 1  + (0.404 − 0.914i)2-s + (0.0320 − 0.999i)3-s + (−0.672 − 0.740i)4-s + (−0.572 − 0.820i)5-s + (−0.900 − 0.433i)6-s + (0.404 + 0.914i)7-s + (−0.949 + 0.315i)8-s + (−0.997 − 0.0640i)9-s + (−0.981 + 0.191i)10-s + (−0.462 − 0.886i)11-s + (−0.761 + 0.648i)12-s + (−0.838 + 0.545i)13-s + 14-s + (−0.838 + 0.545i)15-s + (−0.0960 + 0.995i)16-s + (0.284 − 0.958i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(197\)
Sign: $-0.806 + 0.591i$
Analytic conductor: \(0.914864\)
Root analytic conductor: \(0.914864\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{197} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 197,\ (0:\ ),\ -0.806 + 0.591i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2737382901 - 0.8352637745i\)
\(L(\frac12)\) \(\approx\) \(-0.2737382901 - 0.8352637745i\)
\(L(1)\) \(\approx\) \(0.4632928718 - 0.8232607143i\)
\(L(1)\) \(\approx\) \(0.4632928718 - 0.8232607143i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad197 \( 1 \)
good2 \( 1 + (0.404 - 0.914i)T \)
3 \( 1 + (0.0320 - 0.999i)T \)
5 \( 1 + (-0.572 - 0.820i)T \)
7 \( 1 + (0.404 + 0.914i)T \)
11 \( 1 + (-0.462 - 0.886i)T \)
13 \( 1 + (-0.838 + 0.545i)T \)
17 \( 1 + (0.284 - 0.958i)T \)
19 \( 1 + (-0.222 - 0.974i)T \)
23 \( 1 + (0.967 - 0.253i)T \)
29 \( 1 + (-0.761 + 0.648i)T \)
31 \( 1 + (0.801 + 0.598i)T \)
37 \( 1 + (-0.0960 - 0.995i)T \)
41 \( 1 + (0.284 - 0.958i)T \)
43 \( 1 + (-0.462 - 0.886i)T \)
47 \( 1 + (0.718 - 0.695i)T \)
53 \( 1 + (0.159 - 0.987i)T \)
59 \( 1 + (-0.981 - 0.191i)T \)
61 \( 1 + (0.0320 + 0.999i)T \)
67 \( 1 + (0.718 - 0.695i)T \)
71 \( 1 + (-0.997 - 0.0640i)T \)
73 \( 1 + (-0.0960 - 0.995i)T \)
79 \( 1 + (-0.572 + 0.820i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.801 - 0.598i)T \)
97 \( 1 + (0.926 - 0.375i)T \)
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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

−27.2262388077919191945330406075, −26.47466457857842302863503444501, −25.92137859177008224073828937976, −24.75352972119374225236684244020, −23.28663306832350383061741573962, −23.08428811265667555045531379058, −22.079086137559837628705006332751, −21.06365076589060767773264138746, −20.14048632502676975273671278642, −18.81262349551827577964345043609, −17.38734905526479219632427443114, −16.966312035176807066930949512183, −15.64071317500097230616884155606, −14.89727581841617897833442798459, −14.47170994915346027133433687307, −13.087688420725593249851025886572, −11.777910467185447898782324676360, −10.51834748983150811382874605852, −9.767413483750101976422447932761, −8.05422851506812827704100231122, −7.54060508249028232528800519564, −6.15516374422211079320861569423, −4.80212510397496489474499629375, −4.04742548126446239546960377118, −2.94662322175308703299973424536, 0.5871299679338466346324134510, 2.073497585080923551049944867648, 3.10736613085036027982972337367, 4.87477580693729998036250765807, 5.54711969797703854671290027641, 7.20007788704247611152143910072, 8.62363935300146422773520449303, 9.10496665189832961206141825077, 11.032368075725229456741635791880, 11.81454379936981897520600989975, 12.469445053604569767247928840747, 13.39345591718010151994453647696, 14.374315234488885835060017144746, 15.52206825100258465471574559849, 16.94087874285658813298506383764, 18.13128490457559292665411339863, 18.984231730804879318638384837858, 19.55810594480673225481593269882, 20.66598801986353458747296186711, 21.49620394105477803950221114417, 22.62034475819545938968271424776, 23.72228331474287441145619129224, 24.255011073315463110208647070840, 24.989267791417991861958057509838, 26.65950728156773006888919470022

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