↑

Properties

Label 1-389-389.55-r0-0-0
Degree $1$
Conductor $389$
Sign $0.460 - 0.887i$
Analytic cond. $1.80650$
Root an. cond. $1.80650$
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  + (−0.536 − 0.843i)2-s + (0.333 − 0.942i)3-s + (−0.423 + 0.905i)4-s + (−0.884 + 0.466i)5-s + (−0.974 + 0.224i)6-s + (−0.852 + 0.523i)7-s + (0.991 − 0.129i)8-s + (−0.777 − 0.628i)9-s + (0.868 + 0.495i)10-s + (−0.995 + 0.0970i)11-s + (0.712 + 0.701i)12-s + (0.756 + 0.653i)13-s + (0.898 + 0.438i)14-s + (0.145 + 0.989i)15-s + (−0.641 − 0.767i)16-s + (0.868 + 0.495i)17-s + ⋯
L(s)  = 1  + (−0.536 − 0.843i)2-s + (0.333 − 0.942i)3-s + (−0.423 + 0.905i)4-s + (−0.884 + 0.466i)5-s + (−0.974 + 0.224i)6-s + (−0.852 + 0.523i)7-s + (0.991 − 0.129i)8-s + (−0.777 − 0.628i)9-s + (0.868 + 0.495i)10-s + (−0.995 + 0.0970i)11-s + (0.712 + 0.701i)12-s + (0.756 + 0.653i)13-s + (0.898 + 0.438i)14-s + (0.145 + 0.989i)15-s + (−0.641 − 0.767i)16-s + (0.868 + 0.495i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(389\)
Sign: $0.460 - 0.887i$
Analytic conductor: \(1.80650\)
Root analytic conductor: \(1.80650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{389} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 389,\ (0:\ ),\ 0.460 - 0.887i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6181142939 - 0.3759064215i\)
\(L(\frac12)\) \(\approx\) \(0.6181142939 - 0.3759064215i\)
\(L(1)\) \(\approx\) \(0.6189779145 - 0.3117087009i\)
\(L(1)\) \(\approx\) \(0.6189779145 - 0.3117087009i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad389 \( 1 \)
good2 \( 1 + (-0.536 - 0.843i)T \)
3 \( 1 + (0.333 - 0.942i)T \)
5 \( 1 + (-0.884 + 0.466i)T \)
7 \( 1 + (-0.852 + 0.523i)T \)
11 \( 1 + (-0.995 + 0.0970i)T \)
13 \( 1 + (0.756 + 0.653i)T \)
17 \( 1 + (0.868 + 0.495i)T \)
19 \( 1 + (0.665 - 0.746i)T \)
23 \( 1 + (-0.884 - 0.466i)T \)
29 \( 1 + (0.997 + 0.0647i)T \)
31 \( 1 + (0.966 - 0.256i)T \)
37 \( 1 + (0.208 - 0.977i)T \)
41 \( 1 + (0.665 + 0.746i)T \)
43 \( 1 + (0.948 + 0.318i)T \)
47 \( 1 + (0.665 + 0.746i)T \)
53 \( 1 + (0.616 - 0.787i)T \)
59 \( 1 + (-0.689 + 0.724i)T \)
61 \( 1 + (0.756 + 0.653i)T \)
67 \( 1 + (-0.590 - 0.807i)T \)
71 \( 1 + (-0.423 + 0.905i)T \)
73 \( 1 + (-0.641 + 0.767i)T \)
79 \( 1 + (0.966 + 0.256i)T \)
83 \( 1 + (-0.590 + 0.807i)T \)
89 \( 1 + (0.981 - 0.193i)T \)
97 \( 1 + (-0.423 - 0.905i)T \)
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

−24.89641873266396316402516443911, −23.542933989910085104552717077228, −23.14901218894792257483428127336, −22.33657872700241990991699336056, −20.80645898235466413116511359134, −20.25775737385237621379566809262, −19.364998705797477741351514873244, −18.56773189475617771592704358462, −17.320247977834918231078103546479, −16.23941472876319577585986585152, −15.96765106080072340781055397585, −15.37738058718442883228209388340, −14.088290389847839749479735605483, −13.39428214615250665346673034265, −11.99471885452750193055421064712, −10.55970858767556785634385244593, −10.12106364977089251505094789656, −9.06938416184125756763322558920, −8.04367850412956195870832267448, −7.59449993734217121949760838869, −5.99715600890717435238970743797, −5.12419781215446649912989773156, −4.021509458818111591546469860659, −3.076725805267239997287578904966, −0.761094590739918979573063566478, 0.8419198292219024691530679665, 2.468339119279299907955766238319, 3.03725329590346605948428650047, 4.156492535942176252461104975109, 5.99645066597939515722549814016, 7.11965418014448527227632566880, 7.96836841513271813799496578045, 8.70024101887568412115014458137, 9.817034598526491901056633123704, 10.9080508307110077555018345640, 11.88235452745308155818404549147, 12.4626397443129417044755267766, 13.34298075612694356347102546217, 14.28871621962576763117962964109, 15.68677683748268707036659649327, 16.35233488363169721336284672016, 17.87409696561817489348792995390, 18.38230094010158663242773189046, 19.20349791043370459321987311650, 19.56699982224425102307208555948, 20.62017961698782016186111579367, 21.55141494919936848866652042150, 22.73055700483346599149068094160, 23.263871373460677591550810563127, 24.2569433753576959830559020801

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