↑

Properties

Label 1-19e2-361.138-r0-0-0
Degree $1$
Conductor $361$
Sign $0.835 + 0.549i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.515 − 0.856i)2-s + (0.0642 + 0.997i)3-s + (−0.467 − 0.883i)4-s + (0.997 + 0.0734i)5-s + (0.888 + 0.459i)6-s + (−0.926 + 0.376i)7-s + (−0.998 − 0.0550i)8-s + (−0.991 + 0.128i)9-s + (0.577 − 0.816i)10-s + (−0.298 + 0.954i)11-s + (0.851 − 0.523i)12-s + (0.690 + 0.723i)13-s + (−0.155 + 0.987i)14-s + (−0.00918 + 0.999i)15-s + (−0.562 + 0.826i)16-s + (0.999 − 0.0367i)17-s + ⋯
L(s)  = 1  + (0.515 − 0.856i)2-s + (0.0642 + 0.997i)3-s + (−0.467 − 0.883i)4-s + (0.997 + 0.0734i)5-s + (0.888 + 0.459i)6-s + (−0.926 + 0.376i)7-s + (−0.998 − 0.0550i)8-s + (−0.991 + 0.128i)9-s + (0.577 − 0.816i)10-s + (−0.298 + 0.954i)11-s + (0.851 − 0.523i)12-s + (0.690 + 0.723i)13-s + (−0.155 + 0.987i)14-s + (−0.00918 + 0.999i)15-s + (−0.562 + 0.826i)16-s + (0.999 − 0.0367i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $0.835 + 0.549i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ 0.835 + 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.531152802 + 0.4586659425i\)
\(L(\frac12)\) \(\approx\) \(1.531152802 + 0.4586659425i\)
\(L(1)\) \(\approx\) \(1.348179178 + 0.03101984773i\)
\(L(1)\) \(\approx\) \(1.348179178 + 0.03101984773i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.515 - 0.856i)T \)
3 \( 1 + (0.0642 + 0.997i)T \)
5 \( 1 + (0.997 + 0.0734i)T \)
7 \( 1 + (-0.926 + 0.376i)T \)
11 \( 1 + (-0.298 + 0.954i)T \)
13 \( 1 + (0.690 + 0.723i)T \)
17 \( 1 + (0.999 - 0.0367i)T \)
23 \( 1 + (0.832 + 0.554i)T \)
29 \( 1 + (0.209 + 0.977i)T \)
31 \( 1 + (-0.754 + 0.656i)T \)
37 \( 1 + (-0.677 - 0.735i)T \)
41 \( 1 + (0.983 + 0.182i)T \)
43 \( 1 + (-0.995 + 0.0917i)T \)
47 \( 1 + (-0.0459 - 0.998i)T \)
53 \( 1 + (-0.842 + 0.539i)T \)
59 \( 1 + (-0.649 + 0.760i)T \)
61 \( 1 + (0.919 - 0.393i)T \)
67 \( 1 + (0.957 - 0.289i)T \)
71 \( 1 + (0.919 + 0.393i)T \)
73 \( 1 + (-0.531 - 0.847i)T \)
79 \( 1 + (0.418 - 0.908i)T \)
83 \( 1 + (-0.962 - 0.272i)T \)
89 \( 1 + (-0.531 + 0.847i)T \)
97 \( 1 + (0.957 + 0.289i)T \)
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

−24.6878380213779126047609292839, −23.86245427590661850595261698889, −22.96794197118656630811726588004, −22.398693195930706614660671256350, −21.21213652448713734148416968156, −20.44579940908137928882450878284, −18.96873466397386153083915795527, −18.44134655705890856392302234032, −17.30842553147486257297059455147, −16.78065126543705052275874705080, −15.808222217702470738775386300409, −14.48098900478148894303045087396, −13.7478280884477611090147214276, −13.07294134951771105870600806536, −12.59660372907395805833321553346, −11.13531543888816915841991559230, −9.7861120765771375121736459711, −8.67389785898847466582958344022, −7.83747120291440563052955282485, −6.6813036315111198712767467457, −6.03315031683355108790340959032, −5.3420869146858680578542755111, −3.484123802308779911911478168, −2.7126555198084332264210506960, −0.85174602894544593933638933167, 1.68409207976504254926429614557, 2.87399906132039144383340030249, 3.66967291464730498281382124039, 5.00071269055117671475752546109, 5.64803562873027214007844574409, 6.75313952085670338127577779897, 8.898956391737115341654052526612, 9.45770554757033354159212305471, 10.183516163865689936461084350170, 10.964087524145710481157892416926, 12.20950730185567666951724781472, 13.03748037212709904158573203379, 14.03341790317137976407794663748, 14.759075218651449942191951036586, 15.75681682746099023991707186521, 16.696832685596964261797656932539, 17.88268220838663995499138904752, 18.75942138612652423807488735208, 19.792971453432474168366256941066, 20.67171458541097196950273213798, 21.412906766078342701927239805916, 21.85782105145251441267038617113, 22.92947013134887404383978424610, 23.35414352241111845566683102882, 25.06597156103888353117250001551

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