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Properties

Label 1-517-517.150-r1-0-0
Degree $1$
Conductor $517$
Sign $0.511 + 0.859i$
Analytic cond. $55.5593$
Root an. cond. $55.5593$
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  + (−0.598 + 0.800i)2-s + (0.998 − 0.0546i)3-s + (−0.282 − 0.959i)4-s + (−0.484 − 0.874i)5-s + (−0.554 + 0.832i)6-s + (−0.149 − 0.988i)7-s + (0.937 + 0.347i)8-s + (0.994 − 0.109i)9-s + (0.990 + 0.136i)10-s + (−0.334 − 0.942i)12-s + (−0.0954 + 0.995i)13-s + (0.881 + 0.472i)14-s + (−0.531 − 0.847i)15-s + (−0.839 + 0.542i)16-s + (0.230 + 0.973i)17-s + (−0.507 + 0.861i)18-s + ⋯
L(s)  = 1  + (−0.598 + 0.800i)2-s + (0.998 − 0.0546i)3-s + (−0.282 − 0.959i)4-s + (−0.484 − 0.874i)5-s + (−0.554 + 0.832i)6-s + (−0.149 − 0.988i)7-s + (0.937 + 0.347i)8-s + (0.994 − 0.109i)9-s + (0.990 + 0.136i)10-s + (−0.334 − 0.942i)12-s + (−0.0954 + 0.995i)13-s + (0.881 + 0.472i)14-s + (−0.531 − 0.847i)15-s + (−0.839 + 0.542i)16-s + (0.230 + 0.973i)17-s + (−0.507 + 0.861i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.511 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.511 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $0.511 + 0.859i$
Analytic conductor: \(55.5593\)
Root analytic conductor: \(55.5593\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (150, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (1:\ ),\ 0.511 + 0.859i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.518496748 + 0.8628657956i\)
\(L(\frac12)\) \(\approx\) \(1.518496748 + 0.8628657956i\)
\(L(1)\) \(\approx\) \(1.011762997 + 0.2204339355i\)
\(L(1)\) \(\approx\) \(1.011762997 + 0.2204339355i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.598 + 0.800i)T \)
3 \( 1 + (0.998 - 0.0546i)T \)
5 \( 1 + (-0.484 - 0.874i)T \)
7 \( 1 + (-0.149 - 0.988i)T \)
13 \( 1 + (-0.0954 + 0.995i)T \)
17 \( 1 + (0.230 + 0.973i)T \)
19 \( 1 + (0.122 + 0.992i)T \)
23 \( 1 + (-0.576 + 0.816i)T \)
29 \( 1 + (-0.507 + 0.861i)T \)
31 \( 1 + (0.360 - 0.932i)T \)
37 \( 1 + (0.881 - 0.472i)T \)
41 \( 1 + (0.620 + 0.784i)T \)
43 \( 1 + (0.334 - 0.942i)T \)
53 \( 1 + (0.0409 + 0.999i)T \)
59 \( 1 + (0.824 + 0.565i)T \)
61 \( 1 + (0.176 - 0.984i)T \)
67 \( 1 + (0.460 + 0.887i)T \)
71 \( 1 + (0.598 + 0.800i)T \)
73 \( 1 + (0.740 + 0.672i)T \)
79 \( 1 + (0.969 - 0.243i)T \)
83 \( 1 + (-0.758 + 0.652i)T \)
89 \( 1 + (0.682 - 0.730i)T \)
97 \( 1 + (-0.839 - 0.542i)T \)
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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

−22.73563911402757220148689703953, −22.26338621252292619187961222779, −21.36625166365489958086223856721, −20.53679926947880657704727985127, −19.67819144792616500625473966409, −19.14757539556965045887766033018, −18.25665046546218034587106333274, −17.87511504312646098677357839324, −16.1890659036962874612694871403, −15.53426721973503173288650955618, −14.6758190859287744534287009607, −13.6726345973201080008860988377, −12.71944548778838954062785228665, −11.8872305890439837766381464798, −10.94237123588793794224677912623, −9.975137498347649707636091475443, −9.274819974382957427766768264169, −8.262241339930569535942590593166, −7.660370403499516431316390703724, −6.597432228022211461749180093891, −4.872261090477322186001867584544, −3.64103078486219955762355447136, −2.754193960268630294732919667194, −2.354870536818399997236348543347, −0.57424346232263782724788483366, 0.99485607281507558116579303275, 1.83598502802264693617711897915, 3.85873053157046462521481037945, 4.27851284400628699845886719224, 5.709467258227034030081441076047, 6.95754723869852930929793410047, 7.75805791440300392313121062533, 8.31724629037703274718690288545, 9.36760670019767584622815873449, 9.91586333504871211512769368991, 11.11551342135738899634228225187, 12.51552321170444246862008977901, 13.431019903449510244917500687171, 14.19353070642196107509175308178, 14.95570593884768802873135512558, 15.99078026718598291298694331285, 16.56728414060783918132193018381, 17.31408496092826069064461939056, 18.590060557801780589916739210935, 19.29975048429228041232218017386, 19.95956212402733635366779332246, 20.59162907709325660033630428458, 21.65642497098695179143604476735, 23.13023797835292946622990682933, 23.82760348103695896996936839371

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