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Properties

Label 1-675-675.347-r0-0-0
Degree $1$
Conductor $675$
Sign $0.722 + 0.691i$
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  + (−0.970 + 0.241i)2-s + (0.882 − 0.469i)4-s + (0.984 + 0.173i)7-s + (−0.743 + 0.669i)8-s + (−0.961 − 0.275i)11-s + (0.970 + 0.241i)13-s + (−0.997 + 0.0697i)14-s + (0.559 − 0.829i)16-s + (0.207 + 0.978i)17-s + (−0.669 − 0.743i)19-s + (0.999 + 0.0348i)22-s + (0.898 + 0.438i)23-s − 26-s + (0.951 − 0.309i)28-s + (−0.374 + 0.927i)29-s + ⋯
L(s)  = 1  + (−0.970 + 0.241i)2-s + (0.882 − 0.469i)4-s + (0.984 + 0.173i)7-s + (−0.743 + 0.669i)8-s + (−0.961 − 0.275i)11-s + (0.970 + 0.241i)13-s + (−0.997 + 0.0697i)14-s + (0.559 − 0.829i)16-s + (0.207 + 0.978i)17-s + (−0.669 − 0.743i)19-s + (0.999 + 0.0348i)22-s + (0.898 + 0.438i)23-s − 26-s + (0.951 − 0.309i)28-s + (−0.374 + 0.927i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.722 + 0.691i$
Analytic conductor: \(3.13468\)
Root analytic conductor: \(3.13468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (0:\ ),\ 0.722 + 0.691i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9228939603 + 0.3703774541i\)
\(L(\frac12)\) \(\approx\) \(0.9228939603 + 0.3703774541i\)
\(L(1)\) \(\approx\) \(0.7900609554 + 0.1408165204i\)
\(L(1)\) \(\approx\) \(0.7900609554 + 0.1408165204i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.970 + 0.241i)T \)
7 \( 1 + (0.984 + 0.173i)T \)
11 \( 1 + (-0.961 - 0.275i)T \)
13 \( 1 + (0.970 + 0.241i)T \)
17 \( 1 + (0.207 + 0.978i)T \)
19 \( 1 + (-0.669 - 0.743i)T \)
23 \( 1 + (0.898 + 0.438i)T \)
29 \( 1 + (-0.374 + 0.927i)T \)
31 \( 1 + (0.990 + 0.139i)T \)
37 \( 1 + (-0.406 + 0.913i)T \)
41 \( 1 + (0.241 - 0.970i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (-0.139 - 0.990i)T \)
53 \( 1 + (-0.951 + 0.309i)T \)
59 \( 1 + (0.961 - 0.275i)T \)
61 \( 1 + (-0.719 + 0.694i)T \)
67 \( 1 + (0.927 - 0.374i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (-0.406 - 0.913i)T \)
79 \( 1 + (0.374 - 0.927i)T \)
83 \( 1 + (0.529 + 0.848i)T \)
89 \( 1 + (0.913 - 0.406i)T \)
97 \( 1 + (0.788 - 0.615i)T \)
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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

−22.78511120009670684748073351151, −21.33288086231543930660766563338, −20.7774055047784403443165543339, −20.50124154309367515248629031906, −19.11915917832489369139457739862, −18.5586886726925218566189684336, −17.80242673901517319866764639595, −17.13627488884443287168141900249, −16.13218179103121557789002363805, −15.45411078336196423856396546675, −14.51080042718556743134735302457, −13.377097783141019207438647034392, −12.470265639703331798016811355153, −11.448976057787274666442697924689, −10.827974171863129096953053066216, −10.08033374460561358988165738473, −9.00823643198732088059134036355, −8.10198068087038131870795886096, −7.62273556860576478212038078099, −6.46834119802419831886703646312, −5.36549986504994727600984290028, −4.186643616936551082277949654185, −2.91250040687803007998230272452, −1.95855983378564314926239798410, −0.80065963150697361727871612198, 1.116774084535879100366632489301, 2.07273574039541901348646514058, 3.24460188659898259336550953590, 4.77503724366057558096435469896, 5.68440098720252040613741618765, 6.6393805837913650280682272972, 7.69163832236002708475414802419, 8.455746199563981163187738508811, 8.98486624880131307178090851596, 10.36297266987124950521865473103, 10.89485728699387614975637649603, 11.60439391841748608615875545211, 12.8075433963727974870621256906, 13.82726632955401026816833506260, 14.92658888129721500873646595568, 15.44339774641649486467375019976, 16.351746682181423555067615333996, 17.30866418768898311424538456094, 17.853203990348033087885590126628, 18.75774906254617240417538365521, 19.27506549325926872797980594466, 20.47318967511530293733133190442, 21.060562296612946818246393090472, 21.6685041941340695950438578807, 23.27906499811048375226802097137

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