↑

Properties

Label 1-120-120.77-r0-0-0
Degree $1$
Conductor $120$
Sign $0.850 + 0.525i$
Analytic cond. $0.557277$
Root an. cond. $0.557277$
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  + i·7-s + 11-s + i·13-s − i·17-s + 19-s + i·23-s − 29-s + 31-s − i·37-s − 41-s + i·43-s − i·47-s − 49-s − i·53-s − 59-s + ⋯
L(s)  = 1  + i·7-s + 11-s + i·13-s − i·17-s + 19-s + i·23-s − 29-s + 31-s − i·37-s − 41-s + i·43-s − i·47-s − 49-s − i·53-s − 59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(0.557277\)
Root analytic conductor: \(0.557277\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 120,\ (0:\ ),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.045018494 + 0.2968678545i\)
\(L(\frac12)\) \(\approx\) \(1.045018494 + 0.2968678545i\)
\(L(1)\) \(\approx\) \(1.067755273 + 0.1537254296i\)
\(L(1)\) \(\approx\) \(1.067755273 + 0.1537254296i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 + iT \)
23 \( 1 \)
29 \( 1 \)
31 \( 1 \)
37 \( 1 + T \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 - iT \)
67 \( 1 \)
71 \( 1 + T \)
73 \( 1 \)
79 \( 1 \)
83 \( 1 \)
89 \( 1 + iT \)
97 \( 1 \)
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

−29.06216322057709143384076406852, −27.90542305436922262378393633872, −26.98069565206852312656863322519, −26.08320466233313755767664312478, −24.892805319992691979295557536989, −23.996517127682210731882871689532, −22.80539038891467289178604644182, −22.09037676764445448979770736122, −20.55865810889494203164602518798, −19.97445794285182853470320953435, −18.767034412812321602956545955450, −17.412177024356941860575533342440, −16.7859455040690089526217590038, −15.39815716909531008674000481237, −14.302642154272943249016351278436, −13.28571538747068002650639196003, −12.09166054870580722885132376197, −10.79829033351023394284845198614, −9.862422950971996980250644030631, −8.42073359221080999418596614764, −7.24898888258588345178331535884, −6.03156609326013078747956958947, −4.45327060966820957279146239410, −3.26802774942487382303832203273, −1.24914794308171914838859435745, 1.771524504100283521012114101524, 3.34118229368531035512727057786, 4.89211637335378705887066924134, 6.162893892106083111672750170350, 7.404969801292025163358441978329, 8.97102567402891389548682869222, 9.605140183129745729072940535414, 11.50447748325582779647669271860, 11.95534882993288751927868174807, 13.52722550604910622464487896002, 14.52823677395823186089565398153, 15.67677466895013545154732010023, 16.672307223270681461699085849692, 17.9424011129031141401331596994, 18.87502794152144470201297578196, 19.879676972333379992979162381012, 21.15788006995576482659906173301, 22.03923039809707115932912553796, 22.95863975761160064807169262157, 24.385941907922178663609228819748, 24.97412482763540103565326432565, 26.14550833224938302912234712205, 27.24935476209046935113703551560, 28.18096430639544906858734714526, 29.08890362381721308535411616160

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