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Properties

Label 1-199-199.10-r0-0-0
Degree $1$
Conductor $199$
Sign $0.952 + 0.303i$
Analytic cond. $0.924152$
Root an. cond. $0.924152$
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  + (−0.701 − 0.712i)2-s + (0.110 + 0.993i)3-s + (−0.0158 + 0.999i)4-s + (0.928 + 0.371i)5-s + (0.630 − 0.776i)6-s + (0.997 − 0.0634i)7-s + (0.723 − 0.690i)8-s + (−0.975 + 0.220i)9-s + (−0.386 − 0.922i)10-s + (0.841 − 0.540i)11-s + (−0.995 + 0.0950i)12-s + (0.967 − 0.251i)13-s + (−0.745 − 0.666i)14-s + (−0.266 + 0.963i)15-s + (−0.999 − 0.0317i)16-s + (−0.888 − 0.458i)17-s + ⋯
L(s)  = 1  + (−0.701 − 0.712i)2-s + (0.110 + 0.993i)3-s + (−0.0158 + 0.999i)4-s + (0.928 + 0.371i)5-s + (0.630 − 0.776i)6-s + (0.997 − 0.0634i)7-s + (0.723 − 0.690i)8-s + (−0.975 + 0.220i)9-s + (−0.386 − 0.922i)10-s + (0.841 − 0.540i)11-s + (−0.995 + 0.0950i)12-s + (0.967 − 0.251i)13-s + (−0.745 − 0.666i)14-s + (−0.266 + 0.963i)15-s + (−0.999 − 0.0317i)16-s + (−0.888 − 0.458i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.952 + 0.303i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.952 + 0.303i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.099843955 + 0.1709097620i\)
\(L(\frac12)\) \(\approx\) \(1.099843955 + 0.1709097620i\)
\(L(1)\) \(\approx\) \(1.002140344 + 0.06287527459i\)
\(L(1)\) \(\approx\) \(1.002140344 + 0.06287527459i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (-0.701 - 0.712i)T \)
3 \( 1 + (0.110 + 0.993i)T \)
5 \( 1 + (0.928 + 0.371i)T \)
7 \( 1 + (0.997 - 0.0634i)T \)
11 \( 1 + (0.841 - 0.540i)T \)
13 \( 1 + (0.967 - 0.251i)T \)
17 \( 1 + (-0.888 - 0.458i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (-0.857 + 0.513i)T \)
29 \( 1 + (-0.0792 + 0.996i)T \)
31 \( 1 + (0.805 + 0.592i)T \)
37 \( 1 + (-0.939 - 0.342i)T \)
41 \( 1 + (0.296 - 0.954i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.553 - 0.832i)T \)
53 \( 1 + (-0.266 - 0.963i)T \)
59 \( 1 + (-0.327 + 0.945i)T \)
61 \( 1 + (-0.142 + 0.989i)T \)
67 \( 1 + (-0.786 + 0.618i)T \)
71 \( 1 + (-0.745 + 0.666i)T \)
73 \( 1 + (0.873 + 0.486i)T \)
79 \( 1 + (0.472 + 0.881i)T \)
83 \( 1 + (-0.995 - 0.0950i)T \)
89 \( 1 + (-0.823 - 0.567i)T \)
97 \( 1 + (-0.916 + 0.400i)T \)
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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

−26.62399820752463283830188104782, −25.74185160136810597289330204563, −24.80009517087162100512660132969, −24.515890556114911006050094133007, −23.49125616543503307500972199793, −22.41734439559878040045211443862, −20.82196778508122416643688766062, −20.1215985048199747214529138214, −18.877306426084828011695169940375, −18.02879583494046273307400353439, −17.45299825010585137128273407976, −16.67494688670824056927521797298, −15.1492770458129339833279034079, −14.140161505882715397412068774416, −13.58730763751659219227822266216, −12.13481929854748147687839801601, −11.00719627653505888090504097364, −9.665912728772112070801804551735, −8.59386946125578758589281531205, −7.96004912107422932126302074132, −6.491821221744410586186627773874, −5.99361016382138338278625611596, −4.54094030646927159604612562184, −1.95439113364754470587512675251, −1.41697376788878886406922974579, 1.529088194552747239688642156498, 2.86065972742409882843077758289, 4.02039118670065488724942700124, 5.318555562077014764074645185, 6.82645727731135978930832442289, 8.52629636623129566744024610737, 9.024992477570782025316457310561, 10.20004834416991475703419307652, 11.018676105857906335451670771783, 11.652349062152191895542465054628, 13.50596092019180249063078178630, 14.13407835300651944660772513706, 15.509303574489698025558847404098, 16.56785782141041896543737985859, 17.65627537452582932160217462921, 18.01868763061447371675956332102, 19.550186373390978947504900892336, 20.419523968542531180899771528368, 21.26454820197318606747547406720, 21.85005273540618995346378898613, 22.65854213482089025549517757398, 24.43011822453696395551636325635, 25.47543312872044674060796233851, 26.25209855495795588727223855922, 27.05639527665455126303870341872

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