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Properties

Label 1-203-203.3-r0-0-0
Degree $1$
Conductor $203$
Sign $0.956 - 0.292i$
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  + (−0.997 − 0.0747i)2-s + (0.930 + 0.365i)3-s + (0.988 + 0.149i)4-s + (0.0747 − 0.997i)5-s + (−0.900 − 0.433i)6-s + (−0.974 − 0.222i)8-s + (0.733 + 0.680i)9-s + (−0.149 + 0.988i)10-s + (0.680 + 0.733i)11-s + (0.866 + 0.5i)12-s + (−0.222 − 0.974i)13-s + (0.433 − 0.900i)15-s + (0.955 + 0.294i)16-s + (0.866 − 0.5i)17-s + (−0.680 − 0.733i)18-s + (−0.930 + 0.365i)19-s + ⋯
L(s)  = 1  + (−0.997 − 0.0747i)2-s + (0.930 + 0.365i)3-s + (0.988 + 0.149i)4-s + (0.0747 − 0.997i)5-s + (−0.900 − 0.433i)6-s + (−0.974 − 0.222i)8-s + (0.733 + 0.680i)9-s + (−0.149 + 0.988i)10-s + (0.680 + 0.733i)11-s + (0.866 + 0.5i)12-s + (−0.222 − 0.974i)13-s + (0.433 − 0.900i)15-s + (0.955 + 0.294i)16-s + (0.866 − 0.5i)17-s + (−0.680 − 0.733i)18-s + (−0.930 + 0.365i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $0.956 - 0.292i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ 0.956 - 0.292i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.086662455 - 0.1625462298i\)
\(L(\frac12)\) \(\approx\) \(1.086662455 - 0.1625462298i\)
\(L(1)\) \(\approx\) \(0.9903665179 - 0.07287137825i\)
\(L(1)\) \(\approx\) \(0.9903665179 - 0.07287137825i\)

Euler product

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

−26.88270950083389340549716892169, −25.80013269865054472934103201475, −25.50319564100907369312605860691, −24.27531269671009646910687129644, −23.51487247172889114176382193836, −21.76837805322835583388603498012, −21.140504811368028384814766976621, −19.80701252815920308919760064309, −19.02242324668143220939313397943, −18.72004211514909973342353652535, −17.43259711359458765121108812399, −16.47684368367055214321479861002, −15.028303794883444008915578406890, −14.63579965887467820239058051766, −13.43850660126913840498854895111, −11.9303419464524333443987300767, −10.984256397826883826739781046230, −9.798650207612604372488774096, −8.97446107070285188230336627242, −7.90846306504046717701922523548, −6.93220494610301416621418646839, −6.16931828299718351386955055402, −3.73720748641355281351587331861, −2.671142319677985399454864562, −1.51382110499720942609594571598, 1.26850112303596735570397819647, 2.536233640770321840246145462821, 3.93455815294121261358602147997, 5.3554891188734489740433459350, 7.07026582342059656905379921953, 8.0827626325414268523764110325, 8.90939273219889347624956702171, 9.70920910831447853144932488719, 10.60269414129135955123520999260, 12.19666198633474233950324839504, 12.90846821894652817011522079736, 14.49894675901594612976332269513, 15.30468829415510891907134865367, 16.39087133688880176304887861960, 17.08599923657126028680420663413, 18.25647128709513148703440260280, 19.43103086168402111005072022439, 20.07853081553671846053608676013, 20.77448765725302771656684767876, 21.567012966063386194324525562910, 23.139603195745023197058516174603, 24.603938855325064512403030808344, 25.12591802051483490106953629118, 25.687966265454774920071085735040, 27.057640967669577764829629370336

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