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Properties

Label 1-203-203.158-r0-0-0
Degree $1$
Conductor $203$
Sign $0.436 - 0.899i$
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  + (0.988 − 0.149i)2-s + (0.733 − 0.680i)3-s + (0.955 − 0.294i)4-s + (−0.988 + 0.149i)5-s + (0.623 − 0.781i)6-s + (0.900 − 0.433i)8-s + (0.0747 − 0.997i)9-s + (−0.955 + 0.294i)10-s + (−0.0747 − 0.997i)11-s + (0.5 − 0.866i)12-s + (−0.900 − 0.433i)13-s + (−0.623 + 0.781i)15-s + (0.826 − 0.563i)16-s + (0.5 + 0.866i)17-s + (−0.0747 − 0.997i)18-s + (0.733 + 0.680i)19-s + ⋯
L(s)  = 1  + (0.988 − 0.149i)2-s + (0.733 − 0.680i)3-s + (0.955 − 0.294i)4-s + (−0.988 + 0.149i)5-s + (0.623 − 0.781i)6-s + (0.900 − 0.433i)8-s + (0.0747 − 0.997i)9-s + (−0.955 + 0.294i)10-s + (−0.0747 − 0.997i)11-s + (0.5 − 0.866i)12-s + (−0.900 − 0.433i)13-s + (−0.623 + 0.781i)15-s + (0.826 − 0.563i)16-s + (0.5 + 0.866i)17-s + (−0.0747 − 0.997i)18-s + (0.733 + 0.680i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $0.436 - 0.899i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ 0.436 - 0.899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.929574034 - 1.208086744i\)
\(L(\frac12)\) \(\approx\) \(1.929574034 - 1.208086744i\)
\(L(1)\) \(\approx\) \(1.825261121 - 0.6935011005i\)
\(L(1)\) \(\approx\) \(1.825261121 - 0.6935011005i\)

Euler product

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

−26.80755258577040822316255615468, −26.05871970027386987334877871156, −24.94091282990108122259048355188, −24.25280142682906946230757412323, −23.042889311578521984277056858589, −22.39852444691366110427969797204, −21.360630686169329849139039417176, −20.23638620308982098421359913468, −20.02145348567270551539939583206, −18.74387494832238644157732797155, −16.89607231449837328380765980830, −16.11216644478577790490755309726, −15.1770365742452279402884309241, −14.65735030692631966404829182238, −13.548298634619775091073373817931, −12.36418875618800097157678396054, −11.547329366869164527538180039537, −10.30394824506459028868995710907, −9.08002132957138127893286793070, −7.65810229235053763769673355979, −7.113835913616248601036734892149, −5.06211979858294981981643071924, −4.50415873513955658333277618718, −3.35712627389267858413669147401, −2.28800379594680367053615483192, 1.41041296963226868993147520134, 3.10525072989595612358127302600, 3.53997667212891179538535739415, 5.14884272941287119331406719892, 6.475815580348736725638415445432, 7.5641719322454072448881249555, 8.269685119664882058113370892926, 9.9899638164428331865376668906, 11.33258544427077144591167251996, 12.1716179074982495580349753417, 12.99141388705042341151410717906, 14.0864839267903638384769214872, 14.83736541990124703373990564777, 15.67729935020723410881105834368, 16.83891009624331580794824425413, 18.499305339603744762050891516647, 19.41832200250598298163486634824, 19.870210266229821767127135964208, 20.97889066439030966986752298763, 21.97961972088956107846954519988, 23.11769836722016810612066001246, 23.85804900830646935346715936580, 24.52714770453897352751180049052, 25.45062995885326145758541003079, 26.58448848716882523561630195776

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