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Properties

Label 1-28-28.27-r0-0-0
Degree $1$
Conductor $28$
Sign $1$
Analytic cond. $0.130031$
Root an. cond. $0.130031$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s − 11-s − 13-s − 15-s − 17-s + 19-s − 23-s + 25-s + 27-s + 29-s + 31-s − 33-s + 37-s − 39-s − 41-s − 43-s − 45-s + 47-s − 51-s + 53-s + 55-s + 57-s + 59-s − 61-s + 65-s + ⋯
L(s)  = 1  + 3-s − 5-s + 9-s − 11-s − 13-s − 15-s − 17-s + 19-s − 23-s + 25-s + 27-s + 29-s + 31-s − 33-s + 37-s − 39-s − 41-s − 43-s − 45-s + 47-s − 51-s + 53-s + 55-s + 57-s + 59-s − 61-s + 65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(0.130031\)
Root analytic conductor: \(0.130031\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{28} (27, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 28,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8223610378\)
\(L(\frac12)\) \(\approx\) \(0.8223610378\)
\(L(1)\) \(\approx\) \(1.046454884\)
\(L(1)\) \(\approx\) \(1.046454884\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + T \)
5 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 - T \)
97 \( 1 - T \)
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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

−37.512179947842773084657930490390, −36.36045602124227699396162943688, −35.268334030341376930207529526020, −33.849591836732886972765039184334, −32.184175727883333033755618747177, −31.36586297616383901440906943622, −30.39449098988413064026229502805, −28.72073200609003118776449308629, −27.02957442259405572955262877191, −26.38179066809542195016806658376, −24.73886680489813285047033853957, −23.691534626557196405383352007920, −21.9888981476056831334669883160, −20.39328285450433247152743984794, −19.538616099899411451627397917207, −18.20042527549262445931043231016, −16.01626147715059280892693066920, −15.06829496191904340653215529897, −13.544286371829787414922340082154, −12.02206203818595380135277821042, −10.116306854731297791099024344636, −8.39512407283755124263059308174, −7.29088506700227937478527206798, −4.526774087015476109286965871473, −2.77728324207659233094209284012, 2.77728324207659233094209284012, 4.526774087015476109286965871473, 7.29088506700227937478527206798, 8.39512407283755124263059308174, 10.116306854731297791099024344636, 12.02206203818595380135277821042, 13.544286371829787414922340082154, 15.06829496191904340653215529897, 16.01626147715059280892693066920, 18.20042527549262445931043231016, 19.538616099899411451627397917207, 20.39328285450433247152743984794, 21.9888981476056831334669883160, 23.691534626557196405383352007920, 24.73886680489813285047033853957, 26.38179066809542195016806658376, 27.02957442259405572955262877191, 28.72073200609003118776449308629, 30.39449098988413064026229502805, 31.36586297616383901440906943622, 32.184175727883333033755618747177, 33.849591836732886972765039184334, 35.268334030341376930207529526020, 36.36045602124227699396162943688, 37.512179947842773084657930490390

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