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Properties

Label 1-501-501.140-r0-0-0
Degree $1$
Conductor $501$
Sign $0.797 + 0.602i$
Analytic cond. $2.32663$
Root an. cond. $2.32663$
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  + (−0.954 + 0.298i)2-s + (0.822 − 0.569i)4-s + (−0.316 − 0.948i)5-s + (−0.965 + 0.261i)7-s + (−0.614 + 0.788i)8-s + (0.584 + 0.811i)10-s + (0.914 + 0.404i)11-s + (−0.988 + 0.150i)13-s + (0.843 − 0.537i)14-s + (0.351 − 0.936i)16-s + (0.974 + 0.225i)17-s + (−0.982 − 0.188i)19-s + (−0.800 − 0.599i)20-s + (−0.993 − 0.113i)22-s + (0.421 + 0.906i)23-s + ⋯
L(s)  = 1  + (−0.954 + 0.298i)2-s + (0.822 − 0.569i)4-s + (−0.316 − 0.948i)5-s + (−0.965 + 0.261i)7-s + (−0.614 + 0.788i)8-s + (0.584 + 0.811i)10-s + (0.914 + 0.404i)11-s + (−0.988 + 0.150i)13-s + (0.843 − 0.537i)14-s + (0.351 − 0.936i)16-s + (0.974 + 0.225i)17-s + (−0.982 − 0.188i)19-s + (−0.800 − 0.599i)20-s + (−0.993 − 0.113i)22-s + (0.421 + 0.906i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 501 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.797 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 501 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.797 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(501\)    =    \(3 \cdot 167\)
Sign: $0.797 + 0.602i$
Analytic conductor: \(2.32663\)
Root analytic conductor: \(2.32663\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{501} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 501,\ (0:\ ),\ 0.797 + 0.602i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6114981914 + 0.2050239093i\)
\(L(\frac12)\) \(\approx\) \(0.6114981914 + 0.2050239093i\)
\(L(1)\) \(\approx\) \(0.6143074906 + 0.05476476424i\)
\(L(1)\) \(\approx\) \(0.6143074906 + 0.05476476424i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
167 \( 1 \)
good2 \( 1 + (-0.954 + 0.298i)T \)
5 \( 1 + (-0.316 - 0.948i)T \)
7 \( 1 + (-0.965 + 0.261i)T \)
11 \( 1 + (0.914 + 0.404i)T \)
13 \( 1 + (-0.988 + 0.150i)T \)
17 \( 1 + (0.974 + 0.225i)T \)
19 \( 1 + (-0.982 - 0.188i)T \)
23 \( 1 + (0.421 + 0.906i)T \)
29 \( 1 + (-0.421 + 0.906i)T \)
31 \( 1 + (0.776 - 0.629i)T \)
37 \( 1 + (0.700 + 0.713i)T \)
41 \( 1 + (0.206 - 0.978i)T \)
43 \( 1 + (-0.280 + 0.959i)T \)
47 \( 1 + (-0.0567 - 0.998i)T \)
53 \( 1 + (0.489 + 0.872i)T \)
59 \( 1 + (0.974 - 0.225i)T \)
61 \( 1 + (0.726 + 0.686i)T \)
67 \( 1 + (0.316 - 0.948i)T \)
71 \( 1 + (-0.942 + 0.334i)T \)
73 \( 1 + (-0.351 - 0.936i)T \)
79 \( 1 + (0.881 - 0.472i)T \)
83 \( 1 + (0.954 + 0.298i)T \)
89 \( 1 + (0.752 + 0.658i)T \)
97 \( 1 + (0.776 + 0.629i)T \)
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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

−23.45948498887256975063960926026, −22.56235599501612249999201863386, −21.91098996335543010966220480271, −20.93078735101913566619076253997, −19.79106032473058702888182589017, −19.191381703614243877545261562831, −18.83452771571640400226027428247, −17.60496364594690074854273961035, −16.824975953803875690102753692685, −16.15365061886087091385043921996, −15.02471271699069038017994170518, −14.336594032608217967577241454827, −12.91498707318176601369270970906, −12.07075308315269626882677990446, −11.24607584960095179734324453726, −10.213591774147371126183370129015, −9.76904502350467389048500709276, −8.610353933386820403988537981799, −7.55777327494540045181541056599, −6.76952493408455472815176560044, −6.08189799167231787780059110274, −4.08730287772020788275259802324, −3.17130271127985253645593883109, −2.34110827059772222275197726053, −0.61943143932369286760622128585, 0.95217735480753492378480592977, 2.1891713438495335633922071477, 3.58798895581739899074360726847, 4.9298728879948255563138063235, 5.98606829896588139632600685708, 6.95373436700002799635393695842, 7.84052683712257966440946993700, 8.94664635995063301249470076929, 9.464599982387976778108364602844, 10.2753549438798653907918558426, 11.706882881989893074992159772035, 12.22819332814823508871529964201, 13.219954634119950485315656628003, 14.704416136981340598014795664010, 15.275833436254124808379649902477, 16.39320141185987378959456048892, 16.851770497973205185979643925753, 17.52630178993799876815403209392, 18.90978480216051625334716625531, 19.43792996022511313035659850315, 20.00006183303447965982913739743, 21.008697987631083738035772961015, 21.99177055748443199290595888771, 23.1740453434848689301990181157, 23.872595118606466778705882844730

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