Properties

Label 1-283-283.107-r1-0-0
Degree $1$
Conductor $283$
Sign $-0.286 + 0.957i$
Analytic cond. $30.4125$
Root an. cond. $30.4125$
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  + (0.296 − 0.955i)2-s + (−0.274 + 0.961i)3-s + (−0.824 − 0.565i)4-s + (−0.882 − 0.470i)5-s + (0.836 + 0.547i)6-s + (0.441 − 0.897i)7-s + (−0.784 + 0.619i)8-s + (−0.848 − 0.528i)9-s + (−0.711 + 0.703i)10-s + (−0.0779 − 0.996i)11-s + (0.770 − 0.637i)12-s + (0.0556 − 0.998i)13-s + (−0.726 − 0.687i)14-s + (0.695 − 0.718i)15-s + (0.359 + 0.933i)16-s + (0.958 + 0.285i)17-s + ⋯
L(s)  = 1  + (0.296 − 0.955i)2-s + (−0.274 + 0.961i)3-s + (−0.824 − 0.565i)4-s + (−0.882 − 0.470i)5-s + (0.836 + 0.547i)6-s + (0.441 − 0.897i)7-s + (−0.784 + 0.619i)8-s + (−0.848 − 0.528i)9-s + (−0.711 + 0.703i)10-s + (−0.0779 − 0.996i)11-s + (0.770 − 0.637i)12-s + (0.0556 − 0.998i)13-s + (−0.726 − 0.687i)14-s + (0.695 − 0.718i)15-s + (0.359 + 0.933i)16-s + (0.958 + 0.285i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.286 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.286 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.286 + 0.957i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.286 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1573063086 - 0.2113044776i\)
\(L(\frac12)\) \(\approx\) \(-0.1573063086 - 0.2113044776i\)
\(L(1)\) \(\approx\) \(0.6303805569 - 0.3987854240i\)
\(L(1)\) \(\approx\) \(0.6303805569 - 0.3987854240i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.296 - 0.955i)T \)
3 \( 1 + (-0.274 + 0.961i)T \)
5 \( 1 + (-0.882 - 0.470i)T \)
7 \( 1 + (0.441 - 0.897i)T \)
11 \( 1 + (-0.0779 - 0.996i)T \)
13 \( 1 + (0.0556 - 0.998i)T \)
17 \( 1 + (0.958 + 0.285i)T \)
19 \( 1 + (0.892 - 0.451i)T \)
23 \( 1 + (-0.911 + 0.410i)T \)
29 \( 1 + (-0.944 + 0.328i)T \)
31 \( 1 + (-0.999 - 0.0445i)T \)
37 \( 1 + (-0.662 + 0.749i)T \)
41 \( 1 + (0.144 - 0.989i)T \)
43 \( 1 + (-0.920 - 0.390i)T \)
47 \( 1 + (-0.836 + 0.547i)T \)
53 \( 1 + (-0.359 + 0.933i)T \)
59 \( 1 + (-0.380 + 0.924i)T \)
61 \( 1 + (0.964 - 0.264i)T \)
67 \( 1 + (0.166 - 0.986i)T \)
71 \( 1 + (-0.824 + 0.565i)T \)
73 \( 1 + (0.999 - 0.0445i)T \)
79 \( 1 + (-0.920 + 0.390i)T \)
83 \( 1 + (0.951 + 0.306i)T \)
89 \( 1 + (-0.798 + 0.602i)T \)
97 \( 1 + (0.811 + 0.584i)T \)
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

−25.78240267503586849541446739809, −24.914209532188329427436995506214, −24.16430130944621191107831632547, −23.40036374279215164477733318725, −22.71598667878619561573534597951, −21.914268581370644536534373878519, −20.56890661507361570811210495373, −19.22474117780438428564423570013, −18.35161554471484934300283090449, −18.04459247879786048924982794520, −16.66496663523210952529997540832, −15.91837361977099158829435863942, −14.631244065617596862364362928328, −14.3608002262728392050925348711, −12.89530160275195482935116833500, −12.027101972424358744982501876201, −11.53902580110055771318035437121, −9.66573005732952657080190191787, −8.373365117709715734026265469034, −7.62168985866378890323620075974, −6.90295242474057304649599559683, −5.7978643716648729096233800352, −4.79739363616196998606104875116, −3.41661931965379569016145127677, −1.90901190360228190369498351236, 0.08849065308289740751533074047, 1.10369598892081211975029226055, 3.37023253472373627508337617407, 3.69622632397099320810337552880, 4.97282611210550697920507598365, 5.64863424732293335558355102410, 7.74070005629108016459045027097, 8.66901706644657859373456160352, 9.83778186860850668033823082842, 10.76173370575083329505991618867, 11.36988816195667923535823751949, 12.28523275430302284913618010130, 13.49885141119510973321750066617, 14.43943091267182355429884859386, 15.43127703347905971370776800771, 16.42169112988593680024236685675, 17.31787781468898159266307960703, 18.46755221816288757521690498761, 19.711969718220573570270553824777, 20.30114906793851147950080158064, 20.92477637197755904767776506525, 21.97680782413117123237149340838, 22.73878282327152816665844515197, 23.65998559912149337267046338757, 24.18096200955246119677384367790

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