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Properties

Label 1-624-624.275-r1-0-0
Degree $1$
Conductor $624$
Sign $0.629 + 0.777i$
Analytic cond. $67.0581$
Root an. cond. $67.0581$
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  + 5-s + (0.866 − 0.5i)7-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s + 25-s + (0.866 + 0.5i)29-s − i·31-s + (0.866 − 0.5i)35-s + (−0.5 + 0.866i)37-s + (0.866 + 0.5i)41-s + (−0.866 + 0.5i)43-s − i·47-s + (0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + 5-s + (0.866 − 0.5i)7-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s + 25-s + (0.866 + 0.5i)29-s − i·31-s + (0.866 − 0.5i)35-s + (−0.5 + 0.866i)37-s + (0.866 + 0.5i)41-s + (−0.866 + 0.5i)43-s − i·47-s + (0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.629 + 0.777i$
Analytic conductor: \(67.0581\)
Root analytic conductor: \(67.0581\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 624,\ (1:\ ),\ 0.629 + 0.777i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.390990682 + 1.141016222i\)
\(L(\frac12)\) \(\approx\) \(2.390990682 + 1.141016222i\)
\(L(1)\) \(\approx\) \(1.383132027 + 0.1634586546i\)
\(L(1)\) \(\approx\) \(1.383132027 + 0.1634586546i\)

Euler product

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

−22.4042947440756971981315697600, −21.71319859848442015553989937296, −21.190321300525946994463348908802, −20.379025170251161028482133576362, −19.27406675687616201372525454106, −18.34334012729255182823514361691, −17.77130175640076981003046296205, −17.01617795502526019211813846945, −15.98619561689547174034754860378, −15.090429970111062913829679223572, −14.188843705567159196441615302348, −13.51244765274978479340181170665, −12.65447097370098809391468438386, −11.52861633364055196731178360387, −10.76659811384111769251651951535, −9.90998929939378040207737774027, −8.74844372816778043366938045342, −8.29174930386703011374214360623, −6.937187368783040150135724797415, −5.888867708607351056215305397958, −5.29557761760514611983160202703, −4.17307841263050828048566577662, −2.67559190752334766567869265339, −1.99551375124450301182894343119, −0.64903180119696579882965545056, 1.19679509108715568554447338071, 1.99930605369851612068657187713, 3.17751204489384236476553380522, 4.66932963530610926511053476064, 5.16739075059706512829439083330, 6.37568719975979817715604154363, 7.33832251361634672187647738752, 8.18120878271143594304730209378, 9.37570738880920625452831391864, 10.07926796950097892415332185592, 10.8797435105420663651971109738, 11.92659448742350412723046182221, 12.85515445153251528227682846496, 13.912832471805463319028431833450, 14.215971956361853549644598988321, 15.37642145262755597068712230051, 16.29444368800400144531644144540, 17.32480420518384990641118901719, 17.906080838996194326002276578355, 18.41299117647748090102976185899, 19.85969411286091215115670875741, 20.5244470803446642041152083267, 21.16661830718238560748730326522, 21.972928064304568814379581057339, 22.9527329311781252888851322128

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