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Properties

Label 1-5445-5445.344-r1-0-0
Degree $1$
Conductor $5445$
Sign $0.951 - 0.307i$
Analytic cond. $585.146$
Root an. cond. $585.146$
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  + (0.879 − 0.475i)2-s + (0.548 − 0.836i)4-s + (0.948 − 0.318i)7-s + (0.0855 − 0.996i)8-s + (−0.964 − 0.263i)13-s + (0.683 − 0.730i)14-s + (−0.398 − 0.917i)16-s + (−0.985 + 0.170i)17-s + (0.696 + 0.717i)19-s + (0.580 + 0.814i)23-s + (−0.974 + 0.226i)26-s + (0.254 − 0.967i)28-s + (−0.999 + 0.0380i)29-s + (0.161 + 0.986i)31-s + (−0.786 − 0.618i)32-s + ⋯
L(s)  = 1  + (0.879 − 0.475i)2-s + (0.548 − 0.836i)4-s + (0.948 − 0.318i)7-s + (0.0855 − 0.996i)8-s + (−0.964 − 0.263i)13-s + (0.683 − 0.730i)14-s + (−0.398 − 0.917i)16-s + (−0.985 + 0.170i)17-s + (0.696 + 0.717i)19-s + (0.580 + 0.814i)23-s + (−0.974 + 0.226i)26-s + (0.254 − 0.967i)28-s + (−0.999 + 0.0380i)29-s + (0.161 + 0.986i)31-s + (−0.786 − 0.618i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5445\)    =    \(3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $0.951 - 0.307i$
Analytic conductor: \(585.146\)
Root analytic conductor: \(585.146\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5445} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5445,\ (1:\ ),\ 0.951 - 0.307i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.027134500 - 0.6354538449i\)
\(L(\frac12)\) \(\approx\) \(4.027134500 - 0.6354538449i\)
\(L(1)\) \(\approx\) \(1.724141847 - 0.5757846765i\)
\(L(1)\) \(\approx\) \(1.724141847 - 0.5757846765i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.879 - 0.475i)T \)
7 \( 1 + (0.948 - 0.318i)T \)
13 \( 1 + (-0.964 - 0.263i)T \)
17 \( 1 + (-0.985 + 0.170i)T \)
19 \( 1 + (0.696 + 0.717i)T \)
23 \( 1 + (0.580 + 0.814i)T \)
29 \( 1 + (-0.999 + 0.0380i)T \)
31 \( 1 + (0.161 + 0.986i)T \)
37 \( 1 + (0.362 + 0.931i)T \)
41 \( 1 + (-0.380 - 0.924i)T \)
43 \( 1 + (-0.981 - 0.189i)T \)
47 \( 1 + (-0.761 - 0.647i)T \)
53 \( 1 + (0.993 - 0.113i)T \)
59 \( 1 + (0.991 - 0.132i)T \)
61 \( 1 + (-0.851 + 0.524i)T \)
67 \( 1 + (-0.235 + 0.971i)T \)
71 \( 1 + (-0.897 + 0.441i)T \)
73 \( 1 + (0.870 - 0.491i)T \)
79 \( 1 + (0.797 + 0.603i)T \)
83 \( 1 + (0.953 + 0.299i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (-0.483 + 0.875i)T \)
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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

−17.85364779712453517760697875640, −16.81205782423698994397307391301, −16.57229608361062512718744014023, −15.51473624952125131683353935686, −14.994134348080231089118375350931, −14.64420255088976504222538122616, −13.80115431541988976906036374432, −13.229977540322732906871245805921, −12.55951806822489353877898446413, −11.71979920944112970069100965079, −11.367873100063149964468548328683, −10.709341167484113688715783820172, −9.517019336529988781031873248108, −8.92707840561305813073570959287, −8.076720454203605436427958288351, −7.4941411070275583591832374307, −6.84751373985787208015501132714, −6.10415271292674734592854781864, −5.19609151736538076321665575142, −4.76975272790030216081741706948, −4.19544152366537006969259821108, −3.12317533014112587744365161561, −2.37599484646907458230003716680, −1.827565456408358909759730714631, −0.44778661747383918491149618262, 0.73038871974611692980937246836, 1.624862779093542964028306183414, 2.1566836541526201738414709292, 3.16444199016119979447159246704, 3.799389125617736770197531346694, 4.68523387198562424863277551439, 5.15967508645510735779644342981, 5.76513104128518911769451137415, 6.90465052885153453350916589139, 7.25549966705994512750313794271, 8.16791293990293935618260644020, 9.02842381091610594334336190388, 9.94553723385800842159981372123, 10.400340626557346327902949317393, 11.23452549429865858503721561870, 11.70829179907127856268430202748, 12.29552396130221083997390167513, 13.20871068184469320935626664442, 13.62078754265837152754077393406, 14.36324994963897763642749016945, 15.044027038307631725849785765335, 15.30193021252693136042941073855, 16.363183916169240279338268122928, 16.97062893665107734138595727953, 17.80006273545161793767349063863

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