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Properties

Label 1-504-504.11-r0-0-0
Degree $1$
Conductor $504$
Sign $-0.235 - 0.971i$
Analytic cond. $2.34056$
Root an. cond. $2.34056$
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  + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)29-s − 31-s + (0.5 − 0.866i)37-s + (0.5 − 0.866i)41-s + (−0.5 − 0.866i)43-s + 47-s + (−0.5 − 0.866i)53-s − 55-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)29-s − 31-s + (0.5 − 0.866i)37-s + (0.5 − 0.866i)41-s + (−0.5 − 0.866i)43-s + 47-s + (−0.5 − 0.866i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.235 - 0.971i$
Analytic conductor: \(2.34056\)
Root analytic conductor: \(2.34056\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 504,\ (0:\ ),\ -0.235 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6468503032 - 0.8224597862i\)
\(L(\frac12)\) \(\approx\) \(0.6468503032 - 0.8224597862i\)
\(L(1)\) \(\approx\) \(0.8971260485 - 0.3158225862i\)
\(L(1)\) \(\approx\) \(0.8971260485 - 0.3158225862i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 - T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−23.50068070481104743095637011066, −23.27710497648116584470399845152, −22.10889222235625521073574262913, −21.59863218798757122213222570154, −20.29986892314319069539531766139, −19.72699513034906918911551451655, −18.6637600389261980261661111102, −18.15930869419576214761163222434, −17.07726263524590875250540659847, −16.11571395930843040157530593570, −15.25208315055688655227439123809, −14.48872418614656356558978168504, −13.70717667855860158067462413574, −12.5205404766699715977568233795, −11.55742673221748103762118809960, −11.017495433870140262754646230586, −9.79101273543509806487606816982, −9.06132871236985918491998089880, −7.70837649519724464710983242697, −7.02939254046418150269608695697, −6.17880252666678479216697668509, −4.76329333239572096241272765095, −3.8457889313749430620839062630, −2.796719721925294062548857804398, −1.55497272869612895382816468706, 0.5859394572804422432337168656, 1.86022244349538474246171635563, 3.52995208469467615003500899946, 4.10618186907496831281171303155, 5.53042641013758959643815024496, 6.13003241037888532416989927815, 7.68088084822459219346939959859, 8.339593530433422653544545079689, 9.11384930338522191614726333632, 10.36625754798104957166541950865, 11.182068433678392447363266754653, 12.30543946771050540652870382578, 12.80821566847949595058058663774, 13.9028972199876709516656140408, 14.863127890443046032714297758117, 15.79034635967914130111687273964, 16.63188906207255321685892290592, 17.17602378178577616629072214621, 18.457947847507858380954768354265, 19.19473952772313254041872527758, 20.07517837919469650283718352886, 20.7819799406475529697313786349, 21.62541420439431088228696065417, 22.65149856664432949137047334429, 23.45297601886535878501934530500

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