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Properties

Label 1-624-624.395-r1-0-0
Degree $1$
Conductor $624$
Sign $-0.995 + 0.0985i$
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 + i·7-s − 11-s + 17-s + 19-s + 23-s + 25-s + i·29-s + i·31-s − i·35-s − 37-s − i·41-s + i·43-s + i·47-s − 49-s + ⋯
L(s)  = 1  − 5-s + i·7-s − 11-s + 17-s + 19-s + 23-s + 25-s + i·29-s + i·31-s − i·35-s − 37-s − i·41-s + i·43-s + i·47-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.995 + 0.0985i$
Analytic conductor: \(67.0581\)
Root analytic conductor: \(67.0581\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 624,\ (1:\ ),\ -0.995 + 0.0985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02747115613 + 0.5562202563i\)
\(L(\frac12)\) \(\approx\) \(0.02747115613 + 0.5562202563i\)
\(L(1)\) \(\approx\) \(0.7756988636 + 0.1759494862i\)
\(L(1)\) \(\approx\) \(0.7756988636 + 0.1759494862i\)

Euler product

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

−22.8278646173903733410070392037, −21.40474326870149196232607930504, −20.58229062598218805745803992343, −20.04740342031480001753320423407, −18.99408187686124432210650777950, −18.49076504094872386399942908574, −17.23533414099060888616554263812, −16.557868948925002089819530624659, −15.70056782793750521491112570712, −14.9905067087579309489349217151, −13.90903800438744715265705998097, −13.17080203670698688341025801766, −12.17106540977827813974700786539, −11.32971658193806283889136173458, −10.50408089065323883478158606261, −9.6901878809390428780521748740, −8.35968399820101081088537924202, −7.59601880945023623133883311620, −7.073280583715912876340495400720, −5.60891729450391537052332697521, −4.653974699973827619407238908839, −3.680852949762272129107121085772, −2.84507808517563351727835291361, −1.13452275350099166890953158045, −0.15786023546282091907617364100, 1.28445916509778728540685897072, 2.83675867312750567189574562861, 3.41385785710782291707818445987, 4.95429594736398676778760539479, 5.42402298175729537210340519415, 6.84477741140852289249675603626, 7.7264412053081134742897482143, 8.48517625226761065711785718048, 9.39957739465390571936498681853, 10.55332376301486640512012016374, 11.37553417986063754813625716159, 12.304628399497934688457459406855, 12.76379853934665336260831979957, 14.124558754890502463392153452124, 14.92459858815133711980503237491, 15.86280434022208181878520371157, 16.12620529315868956378804665659, 17.46993224169307902651153593496, 18.50569750422816027698194131031, 18.876729865583299211629368538256, 19.82846080101568103705798801302, 20.78387273650301819326837791101, 21.43745113314182584221299521302, 22.50285976806425865370259811946, 23.143252783250535506299311548713

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