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Properties

Label 1-133-133.74-r0-0-0
Degree $1$
Conductor $133$
Sign $0.266 - 0.963i$
Analytic cond. $0.617649$
Root an. cond. $0.617649$
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  + (0.766 − 0.642i)2-s + (0.766 − 0.642i)3-s + (0.173 − 0.984i)4-s + (0.173 + 0.984i)5-s + (0.173 − 0.984i)6-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (0.766 + 0.642i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (0.173 − 0.984i)13-s + (0.766 + 0.642i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + (−0.5 − 0.866i)18-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.766 − 0.642i)3-s + (0.173 − 0.984i)4-s + (0.173 + 0.984i)5-s + (0.173 − 0.984i)6-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (0.766 + 0.642i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (0.173 − 0.984i)13-s + (0.766 + 0.642i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + (−0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.266 - 0.963i$
Analytic conductor: \(0.617649\)
Root analytic conductor: \(0.617649\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (0:\ ),\ 0.266 - 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.549093624 - 1.178741935i\)
\(L(\frac12)\) \(\approx\) \(1.549093624 - 1.178741935i\)
\(L(1)\) \(\approx\) \(1.595249783 - 0.8277091227i\)
\(L(1)\) \(\approx\) \(1.595249783 - 0.8277091227i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 + (0.766 - 0.642i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.939 + 0.342i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (-0.939 - 0.342i)T \)
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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

−28.876590685322318672693362661209, −27.62306501739576710883098134278, −26.500987554931921397896411812319, −25.84602398774195398411207863657, −24.60760317811151644255568837189, −24.21223303965910060240215773735, −22.82855684595763854402995014556, −21.59056268006648153024451904850, −21.02229929044106271454695734043, −20.20481962764821290939629124158, −18.78597994933641703430230848855, −17.14031549324162794117485347486, −16.13658595105361188151159490951, −15.76001030466945646049239544670, −14.12684287280183925406501806560, −13.74646932655578304079484527004, −12.51703467906111228497086162636, −11.21659306845108312114977971306, −9.47777675705318144728731194412, −8.58624965302350917431240112242, −7.61320167719087260697559210509, −5.88631190386147739580806066303, −4.78941276429890706069615710201, −3.8258778380118635156828622243, −2.37240518558274870339907566004, 1.77341879801315726198566797685, 2.82144200717712457157415112697, 3.8670230758086386546562801012, 5.700340963375543502054733998301, 6.82756970115836589600476792487, 8.01038849938924260301687385774, 9.76856792222474209718589383784, 10.54906243350638228663578185880, 11.97679276373434768260443452729, 12.97016049125140930979901135243, 13.80711668376597694525816874139, 14.92608787654811284985637940559, 15.368838220868617926752562560948, 17.73131283753983612171868740089, 18.49171548915543936700612509117, 19.48108930509438178353324567369, 20.33366966158334639540183724343, 21.30547639213617799369638229817, 22.47073503148202071672131772966, 23.29531108357342309648853512813, 24.27429311709281504341103742278, 25.46510592572377381899588034596, 26.138398764596925158431295071120, 27.57509485510611546408477656977, 28.72215764175349319912681183965

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