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Properties

Label 1-517-517.439-r1-0-0
Degree $1$
Conductor $517$
Sign $-0.418 + 0.908i$
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  + (−0.460 − 0.887i)2-s + (−0.334 + 0.942i)3-s + (−0.576 + 0.816i)4-s + (−0.917 − 0.398i)5-s + (0.990 − 0.136i)6-s + (−0.854 − 0.519i)7-s + (0.990 + 0.136i)8-s + (−0.775 − 0.631i)9-s + (0.0682 + 0.997i)10-s + (−0.576 − 0.816i)12-s + (−0.203 − 0.979i)13-s + (−0.0682 + 0.997i)14-s + (0.682 − 0.730i)15-s + (−0.334 − 0.942i)16-s + (−0.962 − 0.269i)17-s + (−0.203 + 0.979i)18-s + ⋯
L(s)  = 1  + (−0.460 − 0.887i)2-s + (−0.334 + 0.942i)3-s + (−0.576 + 0.816i)4-s + (−0.917 − 0.398i)5-s + (0.990 − 0.136i)6-s + (−0.854 − 0.519i)7-s + (0.990 + 0.136i)8-s + (−0.775 − 0.631i)9-s + (0.0682 + 0.997i)10-s + (−0.576 − 0.816i)12-s + (−0.203 − 0.979i)13-s + (−0.0682 + 0.997i)14-s + (0.682 − 0.730i)15-s + (−0.334 − 0.942i)16-s + (−0.962 − 0.269i)17-s + (−0.203 + 0.979i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $-0.418 + 0.908i$
Analytic conductor: \(55.5593\)
Root analytic conductor: \(55.5593\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (1:\ ),\ -0.418 + 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1081826662 - 0.1689677078i\)
\(L(\frac12)\) \(\approx\) \(-0.1081826662 - 0.1689677078i\)
\(L(1)\) \(\approx\) \(0.4296966646 - 0.2001809409i\)
\(L(1)\) \(\approx\) \(0.4296966646 - 0.2001809409i\)

Euler product

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

−24.095299435022680885909185502804, −23.13589762098211431027178079704, −22.68908521660458460994943063490, −21.82391787400652420883271214136, −19.96007295056917434600729876219, −19.3438740432512668989287107730, −18.83210019829980474211602516565, −18.07683758024674113257882640092, −17.154634544868597408512603084864, −16.193841481321196384615957801897, −15.64129971203441515773484652536, −14.60207794275684575333115421456, −13.70896693423004877409469834991, −12.797936638325266131360281366484, −11.76628257179141128301319388164, −11.029999586931571098487748925234, −9.69918792605258866925654506417, −8.78765615291503133936271773786, −7.830413966165519220803025146193, −7.0135690972326171576710920712, −6.44183369280617279463697267141, −5.45710828299972467033017045629, −4.18401894395984132378214575018, −2.743612801079735353638374466269, −1.30639161457749274916640484630, 0.11085158435144602217297676198, 0.669924655021288783008374685530, 2.76955360410878810495216774986, 3.57173904689438784764293273941, 4.37295890457812345318202462494, 5.30196162969522872836307231165, 6.934355185813029314455514452449, 7.95316586943133880563775538087, 9.08019252600456730326317914526, 9.581063191773186647285370413842, 10.857112665732257254795594870680, 11.03401182308391031180089200113, 12.36048091912663637885553777984, 12.842280393457686569253385674, 14.09498252881619770016969617824, 15.43309672398294866530793261168, 16.06409175663790100591858313368, 16.81788710949481457048226412327, 17.62457665408425400356254655290, 18.66797498534352389758819228715, 19.84377104560762300507836944650, 20.11678433858658281941659360796, 20.79383234069031453486814278274, 22.0920130518239435303330078104, 22.54486223583435166706119299783

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