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Properties

Label 1-203-203.25-r0-0-0
Degree $1$
Conductor $203$
Sign $-0.207 - 0.978i$
Analytic cond. $0.942728$
Root an. cond. $0.942728$
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  + (0.826 − 0.563i)2-s + (−0.988 − 0.149i)3-s + (0.365 − 0.930i)4-s + (0.826 − 0.563i)5-s + (−0.900 + 0.433i)6-s + (−0.222 − 0.974i)8-s + (0.955 + 0.294i)9-s + (0.365 − 0.930i)10-s + (0.955 − 0.294i)11-s + (−0.5 + 0.866i)12-s + (−0.222 + 0.974i)13-s + (−0.900 + 0.433i)15-s + (−0.733 − 0.680i)16-s + (−0.5 − 0.866i)17-s + (0.955 − 0.294i)18-s + (−0.988 + 0.149i)19-s + ⋯
L(s)  = 1  + (0.826 − 0.563i)2-s + (−0.988 − 0.149i)3-s + (0.365 − 0.930i)4-s + (0.826 − 0.563i)5-s + (−0.900 + 0.433i)6-s + (−0.222 − 0.974i)8-s + (0.955 + 0.294i)9-s + (0.365 − 0.930i)10-s + (0.955 − 0.294i)11-s + (−0.5 + 0.866i)12-s + (−0.222 + 0.974i)13-s + (−0.900 + 0.433i)15-s + (−0.733 − 0.680i)16-s + (−0.5 − 0.866i)17-s + (0.955 − 0.294i)18-s + (−0.988 + 0.149i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $-0.207 - 0.978i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ -0.207 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9773757845 - 1.206532804i\)
\(L(\frac12)\) \(\approx\) \(0.9773757845 - 1.206532804i\)
\(L(1)\) \(\approx\) \(1.169928848 - 0.7496318328i\)
\(L(1)\) \(\approx\) \(1.169928848 - 0.7496318328i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.826 - 0.563i)T \)
3 \( 1 + (-0.988 - 0.149i)T \)
5 \( 1 + (0.826 - 0.563i)T \)
11 \( 1 + (0.955 - 0.294i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.988 + 0.149i)T \)
23 \( 1 + (0.0747 - 0.997i)T \)
31 \( 1 + (0.0747 + 0.997i)T \)
37 \( 1 + (0.955 + 0.294i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (-0.733 - 0.680i)T \)
53 \( 1 + (0.0747 + 0.997i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.365 + 0.930i)T \)
67 \( 1 + (-0.733 + 0.680i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (0.826 + 0.563i)T \)
79 \( 1 + (0.955 + 0.294i)T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (0.826 - 0.563i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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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

−27.049515503212641829553289894171, −25.92038870962294890965480698952, −25.112488645039568261289438619860, −24.200456841935356192038985466083, −23.16120842310574873077480526587, −22.38967842344237799934250521339, −21.816877721713580165158986177366, −20.99789938940277045624341386471, −19.56443422856916402275649960035, −17.94839690963389086201724234074, −17.387136998915000572726904234609, −16.6890419683997949456919795897, −15.272335452733687729023437063619, −14.77786628583295985609914796816, −13.35923833744188813238481471030, −12.66816675722579975578576901426, −11.47094020672890577019482369835, −10.5895683683025505851199350500, −9.34232298514953767704143267637, −7.656166643681609787359881734082, −6.453953956488712872246365521019, −5.98574389406882550668883239903, −4.79277904240039873188797847912, −3.62140929906609802311685172940, −1.979019024835560553034591772862, 1.135773822182847419804261287689, 2.28145073183131034504013781127, 4.2430691810009905571443067544, 4.95317040153279757810258776196, 6.21971016475937214387959162931, 6.7407775918869033632814350752, 8.98656646973826882133305279933, 9.97090013752238924992695148096, 11.07027178287349206991642253615, 11.9451070845248046423735705311, 12.77707312417546522463140305963, 13.72326503466675496243112956276, 14.655269968357158976937156791894, 16.20063146490122363163302198927, 16.80736107585230347634604596191, 18.00277896748123570129605596537, 19.0060285190865112050807403512, 20.0970373587403681863282264376, 21.297713593512195123313750727709, 21.77255987975801545789090844338, 22.68444851723059233175708455666, 23.63732515263510521756506931142, 24.549704621058220604469623876712, 25.07040851386915583934650631888, 26.86089175750689138810718387003

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