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Properties

Label 1-19e2-361.172-r0-0-0
Degree $1$
Conductor $361$
Sign $0.999 + 0.0174i$
Analytic cond. $1.67647$
Root an. cond. $1.67647$
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  + (0.546 + 0.837i)2-s + (0.945 − 0.324i)3-s + (−0.401 + 0.915i)4-s + (−0.677 − 0.735i)5-s + (0.789 + 0.614i)6-s + (−0.401 − 0.915i)7-s + (−0.986 + 0.164i)8-s + (0.789 − 0.614i)9-s + (0.245 − 0.969i)10-s + (0.789 + 0.614i)11-s + (−0.0825 + 0.996i)12-s + (0.945 − 0.324i)13-s + (0.546 − 0.837i)14-s + (−0.879 − 0.475i)15-s + (−0.677 − 0.735i)16-s + (−0.401 − 0.915i)17-s + ⋯
L(s)  = 1  + (0.546 + 0.837i)2-s + (0.945 − 0.324i)3-s + (−0.401 + 0.915i)4-s + (−0.677 − 0.735i)5-s + (0.789 + 0.614i)6-s + (−0.401 − 0.915i)7-s + (−0.986 + 0.164i)8-s + (0.789 − 0.614i)9-s + (0.245 − 0.969i)10-s + (0.789 + 0.614i)11-s + (−0.0825 + 0.996i)12-s + (0.945 − 0.324i)13-s + (0.546 − 0.837i)14-s + (−0.879 − 0.475i)15-s + (−0.677 − 0.735i)16-s + (−0.401 − 0.915i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $0.999 + 0.0174i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ 0.999 + 0.0174i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.002224663 + 0.01742474540i\)
\(L(\frac12)\) \(\approx\) \(2.002224663 + 0.01742474540i\)
\(L(1)\) \(\approx\) \(1.599244951 + 0.2028786012i\)
\(L(1)\) \(\approx\) \(1.599244951 + 0.2028786012i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.546 + 0.837i)T \)
3 \( 1 + (0.945 - 0.324i)T \)
5 \( 1 + (-0.677 - 0.735i)T \)
7 \( 1 + (-0.401 - 0.915i)T \)
11 \( 1 + (0.789 + 0.614i)T \)
13 \( 1 + (0.945 - 0.324i)T \)
17 \( 1 + (-0.401 - 0.915i)T \)
23 \( 1 + (0.945 + 0.324i)T \)
29 \( 1 + (-0.401 - 0.915i)T \)
31 \( 1 + (0.546 - 0.837i)T \)
37 \( 1 + (0.789 + 0.614i)T \)
41 \( 1 + (-0.879 - 0.475i)T \)
43 \( 1 + (0.245 + 0.969i)T \)
47 \( 1 + (0.789 + 0.614i)T \)
53 \( 1 + (0.789 + 0.614i)T \)
59 \( 1 + (-0.879 - 0.475i)T \)
61 \( 1 + (-0.986 - 0.164i)T \)
67 \( 1 + (-0.986 + 0.164i)T \)
71 \( 1 + (-0.986 + 0.164i)T \)
73 \( 1 + (-0.401 - 0.915i)T \)
79 \( 1 + (0.245 + 0.969i)T \)
83 \( 1 + (-0.677 + 0.735i)T \)
89 \( 1 + (-0.401 + 0.915i)T \)
97 \( 1 + (-0.986 - 0.164i)T \)
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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

−24.68661521575472265324737181196, −23.70901231946419813543349815446, −22.71867947539820965192798859944, −21.81993264228759891392258880634, −21.45552336575400175759939189035, −20.22707908225243613246846912439, −19.455676242387814202792168150815, −18.91658584930781093107121087292, −18.24627201762148325118775619201, −16.35008145799781195869604620147, −15.34763536791976500769147753216, −14.853192470468571656048077320565, −13.960841083153576500076534801373, −13.04395562193140731714359892455, −12.0228970854566278811593216023, −11.05903605013392398469666693602, −10.319086335999956267651693350027, −8.90620025337355788946319465407, −8.6750122442748506476252864228, −6.89530645132600258049936494324, −5.910637528503532564164237336054, −4.38726382028029951923369231204, −3.50485422301752278691083553455, −2.89602872112791834155387151602, −1.634337869084054307625115651171, 1.055200082305549114974304316787, 2.99314914828358599654986921137, 3.99167698651133166401753690636, 4.56535749192062863543465923963, 6.237362570758981506726289892213, 7.235951224584689208146447217609, 7.82906256854654013715334745900, 8.898092262308068752202552544874, 9.584369827658847206883473352078, 11.413201247643855644876741348178, 12.44375095211919692461903778960, 13.353047668253507225978786374703, 13.73614963812489255243758164870, 15.03885162348360110978117885550, 15.57209252645357183268060544210, 16.57897421922257454705427734175, 17.367768705371759049261462127167, 18.55347989366936218325407084360, 19.626285693139035372186979743546, 20.484390218646261559050709310982, 20.89763391297107114899960321786, 22.47986315621951356992509144827, 23.1694470739962695237927719478, 23.88935733196241923917242963480, 24.84011354839774644509830000448

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