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Properties

Label 1-4921-4921.1004-r0-0-0
Degree $1$
Conductor $4921$
Sign $-0.435 + 0.900i$
Analytic cond. $22.8530$
Root an. cond. $22.8530$
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  + (0.342 + 0.939i)2-s + (−0.5 − 0.866i)3-s + (−0.766 + 0.642i)4-s + (0.866 + 0.5i)5-s + (0.642 − 0.766i)6-s + (−0.866 − 0.5i)8-s + (−0.5 + 0.866i)9-s + (−0.173 + 0.984i)10-s − 11-s + (0.939 + 0.342i)12-s + (−0.642 − 0.766i)13-s − i·15-s + (0.173 − 0.984i)16-s + (0.642 − 0.766i)17-s + (−0.984 − 0.173i)18-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)2-s + (−0.5 − 0.866i)3-s + (−0.766 + 0.642i)4-s + (0.866 + 0.5i)5-s + (0.642 − 0.766i)6-s + (−0.866 − 0.5i)8-s + (−0.5 + 0.866i)9-s + (−0.173 + 0.984i)10-s − 11-s + (0.939 + 0.342i)12-s + (−0.642 − 0.766i)13-s − i·15-s + (0.173 − 0.984i)16-s + (0.642 − 0.766i)17-s + (−0.984 − 0.173i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4921 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4921 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4921\)    =    \(7 \cdot 19 \cdot 37\)
Sign: $-0.435 + 0.900i$
Analytic conductor: \(22.8530\)
Root analytic conductor: \(22.8530\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4921} (1004, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4921,\ (0:\ ),\ -0.435 + 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6182913054 + 0.9859919279i\)
\(L(\frac12)\) \(\approx\) \(0.6182913054 + 0.9859919279i\)
\(L(1)\) \(\approx\) \(0.8836043490 + 0.3628091146i\)
\(L(1)\) \(\approx\) \(0.8836043490 + 0.3628091146i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.342 + 0.939i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.642 - 0.766i)T \)
17 \( 1 + (0.642 - 0.766i)T \)
23 \( 1 + (-0.642 + 0.766i)T \)
29 \( 1 + (0.342 + 0.939i)T \)
31 \( 1 + iT \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.342 - 0.939i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.342 + 0.939i)T \)
61 \( 1 + (0.342 - 0.939i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (0.939 + 0.342i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.342 + 0.939i)T \)
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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

−17.84643586210128048081016439553, −17.219239764144575921994743345406, −16.6644105848350214619158907090, −15.9114648634439115065851876931, −15.043346232260113783882365798212, −14.462396965739262613391517313040, −13.81909487533603986973815696601, −13.02232530915513649554096139215, −12.44404200182774888515463685915, −11.826501671982916804357952783759, −11.08615315597903196021013439364, −10.22916972250714097452480416669, −10.03012354550717090459456493519, −9.365504702467856476928151699637, −8.63398473795963177348425041178, −7.82661018876060129779000436809, −6.32159588785721430756618232554, −5.92973085358896274168830732826, −5.21299148471646687427646233868, −4.50003460127411505696796464396, −4.11801481922787080283477706977, −2.92025595779763245942912820270, −2.36921241433057279303331132245, −1.43626394169552206690513323070, −0.37130985199776887247155753946, 0.80037342083466044523948628249, 2.011400508803400646384135817570, 2.804928708427747861748185612, 3.43679407113371799877163672131, 4.935943949669342877502204619989, 5.3405542415191366407281838154, 5.756781482767099226983621244700, 6.66909194025122027702757753487, 7.27293561070712995484207460453, 7.68958582732983991561825102252, 8.52037227485496372322945841443, 9.39500850495115386350919443624, 10.29579290184874457046337166628, 10.69525886873395983923277549932, 12.02510819431290695598308074970, 12.27614338648562632503279946562, 13.23924897187267525365636240928, 13.55163039332216002898921450883, 14.23567402084063585311857248260, 14.82076457363517160961390875788, 15.76371600362297902375747112709, 16.278218888791463486003584615975, 17.10963774149482140803837632403, 17.73031752596560893811900402316, 17.98129068196820836587281829528

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