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Properties

Label 1-389-389.25-r0-0-0
Degree $1$
Conductor $389$
Sign $0.706 - 0.708i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.912 + 0.408i)2-s + (−0.536 + 0.843i)3-s + (0.665 − 0.746i)4-s + (−0.689 + 0.724i)5-s + (0.145 − 0.989i)6-s + (0.991 + 0.129i)7-s + (−0.302 + 0.953i)8-s + (−0.423 − 0.905i)9-s + (0.333 − 0.942i)10-s + (−0.986 + 0.161i)11-s + (0.271 + 0.962i)12-s + (0.616 − 0.787i)13-s + (−0.957 + 0.287i)14-s + (−0.240 − 0.970i)15-s + (−0.113 − 0.993i)16-s + (0.333 − 0.942i)17-s + ⋯
L(s)  = 1  + (−0.912 + 0.408i)2-s + (−0.536 + 0.843i)3-s + (0.665 − 0.746i)4-s + (−0.689 + 0.724i)5-s + (0.145 − 0.989i)6-s + (0.991 + 0.129i)7-s + (−0.302 + 0.953i)8-s + (−0.423 − 0.905i)9-s + (0.333 − 0.942i)10-s + (−0.986 + 0.161i)11-s + (0.271 + 0.962i)12-s + (0.616 − 0.787i)13-s + (−0.957 + 0.287i)14-s + (−0.240 − 0.970i)15-s + (−0.113 − 0.993i)16-s + (0.333 − 0.942i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.706 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.706 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(389\)
Sign: $0.706 - 0.708i$
Analytic conductor: \(1.80650\)
Root analytic conductor: \(1.80650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{389} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 389,\ (0:\ ),\ 0.706 - 0.708i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2813467334 - 0.1167758360i\)
\(L(\frac12)\) \(\approx\) \(0.2813467334 - 0.1167758360i\)
\(L(1)\) \(\approx\) \(0.4502822037 + 0.1496699284i\)
\(L(1)\) \(\approx\) \(0.4502822037 + 0.1496699284i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad389 \( 1 \)
good2 \( 1 + (-0.912 + 0.408i)T \)
3 \( 1 + (-0.536 + 0.843i)T \)
5 \( 1 + (-0.689 + 0.724i)T \)
7 \( 1 + (0.991 + 0.129i)T \)
11 \( 1 + (-0.986 + 0.161i)T \)
13 \( 1 + (0.616 - 0.787i)T \)
17 \( 1 + (0.333 - 0.942i)T \)
19 \( 1 + (-0.937 + 0.348i)T \)
23 \( 1 + (-0.689 - 0.724i)T \)
29 \( 1 + (-0.590 + 0.807i)T \)
31 \( 1 + (-0.816 - 0.577i)T \)
37 \( 1 + (-0.641 - 0.767i)T \)
41 \( 1 + (-0.937 - 0.348i)T \)
43 \( 1 + (0.0161 - 0.999i)T \)
47 \( 1 + (-0.937 - 0.348i)T \)
53 \( 1 + (0.834 + 0.550i)T \)
59 \( 1 + (-0.735 - 0.677i)T \)
61 \( 1 + (0.616 - 0.787i)T \)
67 \( 1 + (0.868 + 0.495i)T \)
71 \( 1 + (0.665 - 0.746i)T \)
73 \( 1 + (-0.113 + 0.993i)T \)
79 \( 1 + (-0.816 + 0.577i)T \)
83 \( 1 + (0.868 - 0.495i)T \)
89 \( 1 + (0.948 - 0.318i)T \)
97 \( 1 + (0.665 + 0.746i)T \)
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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

−24.315921593175510763523099003793, −23.93854717202957781860248590588, −23.217440735306755598127524181684, −21.58100730399698860908497443832, −21.040103969222249225376347072661, −20.00318923282809689466067062558, −19.20271563535936166047801467547, −18.492591982479298663146068153596, −17.63828491607536948635037972715, −16.87721332498581955615986244636, −16.15003784622099049295091459332, −15.0658591334550278535878852082, −13.47722834917709525742076346001, −12.758740331888818356187558669430, −11.74487373437287260030003173185, −11.25119672599563511796359774487, −10.354263655648354649862788286809, −8.75125686384778372669580459173, −8.12634835769173307244368545, −7.51819323247848588909822516833, −6.27475884987580065991394552652, −5.00269673234687567559395091498, −3.76195061349008225375591983360, −2.03984522171769968679687088003, −1.2935991183545063267107006448, 0.27470787049064101836723842780, 2.22319930869624779968014637879, 3.5878641944541682822162514784, 4.988186173312424224638394765893, 5.75891139361652630029894565482, 7.010127676662905251021078850414, 7.981451470659852670508445905250, 8.71746063047579910433002387131, 10.116558188446747519651323271111, 10.72207089118891628710917818243, 11.287334337459281399858476837756, 12.32213516264785418014811662588, 14.22926753065433059478780514491, 14.97789376115128199369559317392, 15.611772343939277399404743661017, 16.33893523152223584110756675994, 17.38509931269277957191540079, 18.36190364459130305925877855894, 18.54552740888207400065359775825, 20.26543032148546609824044276824, 20.57871780339029693671516899591, 21.70137894425657865289266131278, 22.92670261439360598281065752655, 23.426455706848791689175414210767, 24.29497941605262190436822704459

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