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Properties

Label 1-507-507.101-r1-0-0
Degree $1$
Conductor $507$
Sign $-0.337 - 0.941i$
Analytic cond. $54.4847$
Root an. cond. $54.4847$
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  + (0.948 − 0.316i)2-s + (0.799 − 0.600i)4-s + (−0.970 − 0.239i)5-s + (0.996 + 0.0804i)7-s + (0.568 − 0.822i)8-s + (−0.996 + 0.0804i)10-s + (−0.200 − 0.979i)11-s + (0.970 − 0.239i)14-s + (0.278 − 0.960i)16-s + (0.996 + 0.0804i)17-s + (0.5 + 0.866i)19-s + (−0.919 + 0.391i)20-s + (−0.5 − 0.866i)22-s + (0.5 − 0.866i)23-s + (0.885 + 0.464i)25-s + ⋯
L(s)  = 1  + (0.948 − 0.316i)2-s + (0.799 − 0.600i)4-s + (−0.970 − 0.239i)5-s + (0.996 + 0.0804i)7-s + (0.568 − 0.822i)8-s + (−0.996 + 0.0804i)10-s + (−0.200 − 0.979i)11-s + (0.970 − 0.239i)14-s + (0.278 − 0.960i)16-s + (0.996 + 0.0804i)17-s + (0.5 + 0.866i)19-s + (−0.919 + 0.391i)20-s + (−0.5 − 0.866i)22-s + (0.5 − 0.866i)23-s + (0.885 + 0.464i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.337 - 0.941i$
Analytic conductor: \(54.4847\)
Root analytic conductor: \(54.4847\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 507,\ (1:\ ),\ -0.337 - 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.995826304 - 2.834730014i\)
\(L(\frac12)\) \(\approx\) \(1.995826304 - 2.834730014i\)
\(L(1)\) \(\approx\) \(1.667772542 - 0.7928837057i\)
\(L(1)\) \(\approx\) \(1.667772542 - 0.7928837057i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.948 - 0.316i)T \)
5 \( 1 + (-0.970 - 0.239i)T \)
7 \( 1 + (0.996 + 0.0804i)T \)
11 \( 1 + (-0.200 - 0.979i)T \)
17 \( 1 + (0.996 + 0.0804i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.948 + 0.316i)T \)
31 \( 1 + (-0.885 + 0.464i)T \)
37 \( 1 + (0.0402 - 0.999i)T \)
41 \( 1 + (-0.632 - 0.774i)T \)
43 \( 1 + (-0.0402 - 0.999i)T \)
47 \( 1 + (0.120 - 0.992i)T \)
53 \( 1 + (-0.568 + 0.822i)T \)
59 \( 1 + (0.278 + 0.960i)T \)
61 \( 1 + (0.428 - 0.903i)T \)
67 \( 1 + (-0.799 - 0.600i)T \)
71 \( 1 + (0.987 - 0.160i)T \)
73 \( 1 + (0.748 - 0.663i)T \)
79 \( 1 + (0.120 - 0.992i)T \)
83 \( 1 + (-0.354 - 0.935i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.692 - 0.721i)T \)
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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

−23.76155969949794173688573331859, −22.90352215399845958916419253897, −22.192131975513245137057181573423, −21.12684852980233441739765023047, −20.46150549902001272505182400592, −19.73828562502897554738717241383, −18.55809202434661414353549666552, −17.56785713505907801720068304759, −16.69644795375589415173073868373, −15.66731811555981547881904549018, −14.98660439649802429569005579355, −14.478383404628138256382792848882, −13.32729567062384450220649366372, −12.438052704245872839919344241765, −11.4748279847617378419570730954, −11.142743226868674439592145632843, −9.70243136441744005414028830107, −8.17276713833629298021796653983, −7.57403397537651027971498804223, −6.893200471323791131300793001058, −5.40406439602135269231479976949, −4.7258095420613986479159777986, −3.78518612776478677975153525824, −2.753320410009542474195071013300, −1.41417541596760670780786720341, 0.64784777939869690741572130324, 1.79511163641424755598433429155, 3.25682819048426414724512928354, 3.91395789653812248121541471957, 5.10437515450208416551759756331, 5.68154007865898519114354184487, 7.16709064080501069718640660383, 7.92204380814702393436958207040, 8.932560107543597289063250139256, 10.522547801219351771085158538481, 11.07524951219520250966380937873, 12.00099196105998445328989513061, 12.54236181702751829775338056218, 13.780023946791111437247157176423, 14.52316062609462233649251085103, 15.197618012440895794683746192543, 16.26502119843788856699919312673, 16.77972936522248190040832477710, 18.50930760246603286148646936361, 18.91289888169220571504480728015, 20.07262774273011733602315454069, 20.7033927077465244758923737574, 21.38962353995386201172185410217, 22.34887730517930816982993334511, 23.24534325818322128495089739781

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