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Properties

Label 1-1925-1925.1164-r0-0-0
Degree $1$
Conductor $1925$
Sign $0.983 - 0.183i$
Analytic cond. $8.93966$
Root an. cond. $8.93966$
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  + (−0.913 − 0.406i)2-s + (0.104 + 0.994i)3-s + (0.669 + 0.743i)4-s + (0.309 − 0.951i)6-s + (−0.309 − 0.951i)8-s + (−0.978 + 0.207i)9-s + (−0.669 + 0.743i)12-s + (−0.309 + 0.951i)13-s + (−0.104 + 0.994i)16-s + (−0.669 − 0.743i)17-s + (0.978 + 0.207i)18-s + (0.669 − 0.743i)19-s + (0.978 − 0.207i)23-s + (0.913 − 0.406i)24-s + (0.669 − 0.743i)26-s + (−0.309 − 0.951i)27-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)2-s + (0.104 + 0.994i)3-s + (0.669 + 0.743i)4-s + (0.309 − 0.951i)6-s + (−0.309 − 0.951i)8-s + (−0.978 + 0.207i)9-s + (−0.669 + 0.743i)12-s + (−0.309 + 0.951i)13-s + (−0.104 + 0.994i)16-s + (−0.669 − 0.743i)17-s + (0.978 + 0.207i)18-s + (0.669 − 0.743i)19-s + (0.978 − 0.207i)23-s + (0.913 − 0.406i)24-s + (0.669 − 0.743i)26-s + (−0.309 − 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.983 - 0.183i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ 0.983 - 0.183i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8459740023 - 0.07808427441i\)
\(L(\frac12)\) \(\approx\) \(0.8459740023 - 0.07808427441i\)
\(L(1)\) \(\approx\) \(0.6924355165 + 0.07396790851i\)
\(L(1)\) \(\approx\) \(0.6924355165 + 0.07396790851i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.913 - 0.406i)T \)
3 \( 1 + (0.104 + 0.994i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (0.978 - 0.207i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.913 - 0.406i)T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + (-0.913 - 0.406i)T \)
53 \( 1 + (0.978 + 0.207i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 + (0.669 - 0.743i)T \)
67 \( 1 + (0.104 + 0.994i)T \)
71 \( 1 + T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (-0.978 + 0.207i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−19.78439579854778023326940260053, −19.33216055138688557438477397408, −18.34991984418655910173879316414, −18.06653002274925773751120627030, −17.20057668396959484901675036856, −16.733189453128688949873800912436, −15.69063629652318817752779320074, −14.934100039335556546790783997818, −14.38133140817840252665351435050, −13.37200321823312034821045256760, −12.68491219670370318296042216724, −11.84125894576524902160615637054, −11.049776706925559820809752402447, −10.33205372672228988254463581341, −9.39314857638891179720117190309, −8.582046819332035618918340461149, −7.990511545431210703735877353467, −7.26416156822050057100224154258, −6.61123637578889288483799572013, −5.74121870756466443833106036140, −5.11368682729637740538991378195, −3.46994039367194884985127808224, −2.57620655133066280447710739316, −1.65860911386191226718792261836, −0.841894681923158337361353044865, 0.51328539013957328581945634070, 1.958369760445125535030850016052, 2.76324416810129173126201397546, 3.559015981627864641882392627414, 4.5186199122192355011331497196, 5.26035200192728801966037759616, 6.603412922363226697272825311204, 7.16296092891651687311438639187, 8.27367025939615601864736606958, 9.007259626489679775380799450650, 9.46540461928300889948506197379, 10.174829687387580636020038258480, 11.15914470843902882646267168223, 11.41924338367147131769342102027, 12.33340825277129017033128935413, 13.41705680373560972112751210836, 14.19317169715636509397612817841, 15.164257999754184280051093075243, 15.82801048743449775631956348964, 16.38567339606016339240612084994, 17.1150013837528176264125718359, 17.76565171063366627651627234922, 18.56477426336473647409096808623, 19.57021646227991889588753292750, 19.81369317167552122918044761089

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