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Properties

Label 1-675-675.31-r0-0-0
Degree $1$
Conductor $675$
Sign $-0.839 - 0.543i$
Analytic cond. $3.13468$
Root an. cond. $3.13468$
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  + (−0.997 − 0.0697i)2-s + (0.990 + 0.139i)4-s + (0.173 − 0.984i)7-s + (−0.978 − 0.207i)8-s + (0.559 − 0.829i)11-s + (−0.997 + 0.0697i)13-s + (−0.241 + 0.970i)14-s + (0.961 + 0.275i)16-s + (0.669 − 0.743i)17-s + (−0.978 − 0.207i)19-s + (−0.615 + 0.788i)22-s + (−0.719 − 0.694i)23-s + 26-s + (0.309 − 0.951i)28-s + (0.848 − 0.529i)29-s + ⋯
L(s)  = 1  + (−0.997 − 0.0697i)2-s + (0.990 + 0.139i)4-s + (0.173 − 0.984i)7-s + (−0.978 − 0.207i)8-s + (0.559 − 0.829i)11-s + (−0.997 + 0.0697i)13-s + (−0.241 + 0.970i)14-s + (0.961 + 0.275i)16-s + (0.669 − 0.743i)17-s + (−0.978 − 0.207i)19-s + (−0.615 + 0.788i)22-s + (−0.719 − 0.694i)23-s + 26-s + (0.309 − 0.951i)28-s + (0.848 − 0.529i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.839 - 0.543i$
Analytic conductor: \(3.13468\)
Root analytic conductor: \(3.13468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (0:\ ),\ -0.839 - 0.543i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1448892481 - 0.4901849424i\)
\(L(\frac12)\) \(\approx\) \(0.1448892481 - 0.4901849424i\)
\(L(1)\) \(\approx\) \(0.5746274889 - 0.2029013313i\)
\(L(1)\) \(\approx\) \(0.5746274889 - 0.2029013313i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.997 - 0.0697i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (0.559 - 0.829i)T \)
13 \( 1 + (-0.997 + 0.0697i)T \)
17 \( 1 + (0.669 - 0.743i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (-0.719 - 0.694i)T \)
29 \( 1 + (0.848 - 0.529i)T \)
31 \( 1 + (-0.882 + 0.469i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
41 \( 1 + (-0.997 + 0.0697i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.882 - 0.469i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.559 + 0.829i)T \)
61 \( 1 + (0.438 + 0.898i)T \)
67 \( 1 + (0.848 + 0.529i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (-0.104 - 0.994i)T \)
79 \( 1 + (0.848 - 0.529i)T \)
83 \( 1 + (-0.374 + 0.927i)T \)
89 \( 1 + (-0.104 - 0.994i)T \)
97 \( 1 + (0.0348 + 0.999i)T \)
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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

−23.28394609168703959957313316530, −21.91529425723143793835392101625, −21.53597225463265615054790788303, −20.40928212098753834779939347670, −19.63672243302479481819226359527, −19.01917353822762226356133671609, −18.1157724602610403684074552534, −17.39955097561333073580410768235, −16.74548670765750368683899954647, −15.68863570319287232874578236248, −14.90866320081913926690940915023, −14.43847804299168665475418477657, −12.60125931993986385230724647828, −12.21721423818776624119781673079, −11.29850861700786259598124767182, −10.17629139423664302971863192496, −9.58506539271933880051959128767, −8.65401693493051501855843308360, −7.88291575036351218295163247662, −6.92043610800267141567350363611, −6.00397671816019034238000062938, −5.04394782931921135824947969955, −3.59482037694581416665747909915, −2.27052125627108526259676416729, −1.65773072543721923457666265484, 0.3346678306988294699618913544, 1.518979663612819579064617222281, 2.74472444788160609418401298138, 3.79551188987172381129590788770, 5.031864600511931602275400029517, 6.41322986893821124598961463396, 7.02235904002507947415119765055, 8.03274472258917912723147046907, 8.71886621915110157133408969530, 9.9301184073891148472163365572, 10.329154401090675506964085474862, 11.45469181907723343589326445483, 12.02055986188795230759632866016, 13.23936716256081156133061874236, 14.298009218750855338224243603521, 14.95928881483903719670610623390, 16.33840750263636595097415085792, 16.63974986877990983674249244729, 17.450927947926570600792815212906, 18.29821385039868540201019404401, 19.277149237311781059124607251018, 19.76284515296949130693069737969, 20.6036770696350814216540986814, 21.42000293183804096500454564206, 22.26703138024510911502870873940

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