↑

Properties

Label 1-203-203.68-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

Related objects

Downloads

Learn more

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: $\chi_{203} (68, \cdot )$
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 \( 1 + (-0.781 + 0.623i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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