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Properties

Label 1-507-507.122-r0-0-0
Degree $1$
Conductor $507$
Sign $-0.998 + 0.0557i$
Analytic cond. $2.35449$
Root an. cond. $2.35449$
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  + (−0.822 + 0.568i)2-s + (0.354 − 0.935i)4-s + (−0.663 − 0.748i)5-s + (−0.239 − 0.970i)7-s + (0.239 + 0.970i)8-s + (0.970 + 0.239i)10-s + (−0.822 − 0.568i)11-s + (0.748 + 0.663i)14-s + (−0.748 − 0.663i)16-s + (−0.970 + 0.239i)17-s − i·19-s + (−0.935 + 0.354i)20-s + 22-s + 23-s + (−0.120 + 0.992i)25-s + ⋯
L(s)  = 1  + (−0.822 + 0.568i)2-s + (0.354 − 0.935i)4-s + (−0.663 − 0.748i)5-s + (−0.239 − 0.970i)7-s + (0.239 + 0.970i)8-s + (0.970 + 0.239i)10-s + (−0.822 − 0.568i)11-s + (0.748 + 0.663i)14-s + (−0.748 − 0.663i)16-s + (−0.970 + 0.239i)17-s − i·19-s + (−0.935 + 0.354i)20-s + 22-s + 23-s + (−0.120 + 0.992i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.998 + 0.0557i$
Analytic conductor: \(2.35449\)
Root analytic conductor: \(2.35449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 507,\ (0:\ ),\ -0.998 + 0.0557i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.004508505048 - 0.1616462209i\)
\(L(\frac12)\) \(\approx\) \(0.004508505048 - 0.1616462209i\)
\(L(1)\) \(\approx\) \(0.4802176299 - 0.07447746669i\)
\(L(1)\) \(\approx\) \(0.4802176299 - 0.07447746669i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.822 + 0.568i)T \)
5 \( 1 + (-0.663 - 0.748i)T \)
7 \( 1 + (-0.239 - 0.970i)T \)
11 \( 1 + (-0.822 - 0.568i)T \)
17 \( 1 + (-0.970 + 0.239i)T \)
19 \( 1 - iT \)
23 \( 1 + T \)
29 \( 1 + (-0.568 - 0.822i)T \)
31 \( 1 + (0.992 - 0.120i)T \)
37 \( 1 + (-0.992 + 0.120i)T \)
41 \( 1 + (-0.464 + 0.885i)T \)
43 \( 1 + (-0.120 + 0.992i)T \)
47 \( 1 + (-0.935 + 0.354i)T \)
53 \( 1 + (0.970 - 0.239i)T \)
59 \( 1 + (0.663 + 0.748i)T \)
61 \( 1 + (-0.970 - 0.239i)T \)
67 \( 1 + (-0.935 + 0.354i)T \)
71 \( 1 + (-0.464 + 0.885i)T \)
73 \( 1 + (0.822 + 0.568i)T \)
79 \( 1 + (-0.354 - 0.935i)T \)
83 \( 1 + (0.464 + 0.885i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.663 + 0.748i)T \)
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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

−24.16628651097745767876779708847, −22.84768609184448294294722594085, −22.39753068867147896659654308260, −21.35677080652836368971650763866, −20.60611926026688889232742520174, −19.63772559605822638460856762057, −18.84741292194035726517952637970, −18.363146509498977693554950477373, −17.572189056014097792682901850396, −16.35522272866317690592429283146, −15.52888378174882993709400197211, −15.00233593477081847772809121430, −13.50049222547298007048699526927, −12.43399989006997364855752549586, −11.88250217589354457056600595480, −10.84871630214545352873536841781, −10.219661725113634798239396699830, −9.08868871103198502417090680529, −8.30759275400687325251332500805, −7.32236110882470330054346881150, −6.55337234398920994447560084304, −5.06917882188819241594879086428, −3.686371925556808239541499348704, −2.79049292920678452166360945677, −1.92858742900971532576709913467, 0.12184885546166511452914850022, 1.22396451792129622052456134532, 2.85429003065813794837821553619, 4.3471464401271547671914951883, 5.15081758846359137368378046154, 6.450834490697631893796779533489, 7.317079464242340414258191395060, 8.16685602460357472922753100863, 8.89453667356747520161063652003, 9.92352020192203762232942980754, 10.92218993579744250248503420876, 11.52373675231341515337678265950, 13.12397845417098393075173873436, 13.5476650127714579275400390339, 15.009740968531722155785695909390, 15.664059348998487184416141955036, 16.433685858802567522857288285422, 17.128560211281463490310089099429, 17.91234944344613984437492942035, 19.14910524841229736908807976257, 19.578547880951282385081767365348, 20.43995158549290421405208289478, 21.1908296358041807894967366625, 22.77630273881559359811182353366, 23.38256270288549088462327532301

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