Properties

Label 1-517-517.450-r1-0-0
Degree $1$
Conductor $517$
Sign $-0.174 + 0.984i$
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  + (0.990 − 0.136i)2-s + (0.854 + 0.519i)3-s + (0.962 − 0.269i)4-s + (−0.334 + 0.942i)5-s + (0.917 + 0.398i)6-s + (0.0682 + 0.997i)7-s + (0.917 − 0.398i)8-s + (0.460 + 0.887i)9-s + (−0.203 + 0.979i)10-s + (0.962 + 0.269i)12-s + (0.576 − 0.816i)13-s + (0.203 + 0.979i)14-s + (−0.775 + 0.631i)15-s + (0.854 − 0.519i)16-s + (−0.682 + 0.730i)17-s + (0.576 + 0.816i)18-s + ⋯
L(s)  = 1  + (0.990 − 0.136i)2-s + (0.854 + 0.519i)3-s + (0.962 − 0.269i)4-s + (−0.334 + 0.942i)5-s + (0.917 + 0.398i)6-s + (0.0682 + 0.997i)7-s + (0.917 − 0.398i)8-s + (0.460 + 0.887i)9-s + (−0.203 + 0.979i)10-s + (0.962 + 0.269i)12-s + (0.576 − 0.816i)13-s + (0.203 + 0.979i)14-s + (−0.775 + 0.631i)15-s + (0.854 − 0.519i)16-s + (−0.682 + 0.730i)17-s + (0.576 + 0.816i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $-0.174 + 0.984i$
Analytic conductor: \(55.5593\)
Root analytic conductor: \(55.5593\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (450, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (1:\ ),\ -0.174 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.445954671 + 4.110890585i\)
\(L(\frac12)\) \(\approx\) \(3.445954671 + 4.110890585i\)
\(L(1)\) \(\approx\) \(2.348388939 + 1.066252281i\)
\(L(1)\) \(\approx\) \(2.348388939 + 1.066252281i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.990 - 0.136i)T \)
3 \( 1 + (0.854 + 0.519i)T \)
5 \( 1 + (-0.334 + 0.942i)T \)
7 \( 1 + (0.0682 + 0.997i)T \)
13 \( 1 + (0.576 - 0.816i)T \)
17 \( 1 + (-0.682 + 0.730i)T \)
19 \( 1 + (0.334 + 0.942i)T \)
23 \( 1 + (-0.990 - 0.136i)T \)
29 \( 1 + (0.576 + 0.816i)T \)
31 \( 1 + (0.854 - 0.519i)T \)
37 \( 1 + (0.203 - 0.979i)T \)
41 \( 1 + (0.917 + 0.398i)T \)
43 \( 1 + (-0.962 + 0.269i)T \)
53 \( 1 + (-0.917 - 0.398i)T \)
59 \( 1 + (0.962 + 0.269i)T \)
61 \( 1 + (-0.203 - 0.979i)T \)
67 \( 1 + (-0.0682 + 0.997i)T \)
71 \( 1 + (-0.990 - 0.136i)T \)
73 \( 1 + (-0.460 + 0.887i)T \)
79 \( 1 + (0.775 - 0.631i)T \)
83 \( 1 + (-0.682 - 0.730i)T \)
89 \( 1 + (-0.334 + 0.942i)T \)
97 \( 1 + (0.854 + 0.519i)T \)
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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

−23.53940824053972186994318366397, −22.37414380290881261133702990126, −21.12482328319146942228673812842, −20.67883198790970371814696029260, −19.85400265876422762519309084, −19.43947659943505627774124715684, −17.954967477368298032424918431884, −16.985318701426070257165385092486, −15.96843215918497271985493509720, −15.48583763714350465558887108846, −14.13093126284817278969936806785, −13.65083765335034585293363566563, −13.11987226666726916554991851256, −11.99202269795042787032557624335, −11.38765890965445556749747719828, −9.9456602185294144495172432439, −8.7957979349526869596106789675, −7.919796389041448173466219931416, −7.08886499006457790699281890566, −6.268144947870207346610988197802, −4.677397042203178961040038045040, −4.19912929268136475025665289029, −3.120934173166646302038556583377, −1.86883691589453638915200875104, −0.81759661764157575904256708987, 1.83301940653306191094039508567, 2.7376963293447436867891819098, 3.48717064645851175321205525287, 4.35950834924893254967291584511, 5.637450228858454247321874744161, 6.4249100460000181047171125172, 7.74988469103817576339204790019, 8.409895842144090407721425591697, 9.86630142584263112881023385624, 10.59374681524022816951536244947, 11.45447390713239931915502857609, 12.48152900851721890351418833274, 13.374180914640391413187561683197, 14.454088146247574718255417524854, 14.78292388183625787188085623053, 15.7509655923981356909984885614, 16.05627060909404720792113474536, 17.8634824434958373774607539392, 18.79707993969168799492101950139, 19.56296294135441217405056358423, 20.32566627186483063795588584250, 21.236578758865598652048778961321, 21.955694064848000503350005196715, 22.47375974158551437624820851812, 23.393742351117383884009805300363

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