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Properties

Label 1-1925-1925.1154-r0-0-0
Degree $1$
Conductor $1925$
Sign $0.637 + 0.770i$
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  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)6-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (−0.809 + 0.587i)12-s + (0.809 − 0.587i)13-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.951i)17-s + 18-s + (0.309 − 0.951i)19-s + (0.809 + 0.587i)23-s + 24-s − 26-s + (−0.809 − 0.587i)27-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)6-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (−0.809 + 0.587i)12-s + (0.809 − 0.587i)13-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.951i)17-s + 18-s + (0.309 − 0.951i)19-s + (0.809 + 0.587i)23-s + 24-s − 26-s + (−0.809 − 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.637 + 0.770i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ 0.637 + 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.053842264 + 0.4959005275i\)
\(L(\frac12)\) \(\approx\) \(1.053842264 + 0.4959005275i\)
\(L(1)\) \(\approx\) \(0.8335506774 + 0.1254984890i\)
\(L(1)\) \(\approx\) \(0.8335506774 + 0.1254984890i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
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

−19.79631523725669385972161617128, −18.86698061527523736273768079348, −18.43699581128315621589652284665, −18.07280127695692478887923834164, −16.85780451751381800408435437733, −16.62654377593399196838082796554, −15.55742654353778692631352950443, −14.833454959375582977802814750459, −14.03592408396975952454834218353, −13.56619870867180597355964847476, −12.55891068870668264725220782104, −11.62816349695734721344643065012, −11.0599046849280816113135891603, −10.05821601147364935610004415176, −8.97596242934913200027202634394, −8.81412816189226782002616850310, −7.68203690402527744697530977661, −7.24190943263264378306954954158, −6.374605517512228224802071740296, −5.79720980029484803171627196920, −4.743724507089067365435434161, −3.43445383378490697739483420064, −2.37561293035815816842157364480, −1.552764324813964020393020823355, −0.66276897053964411909950295985, 0.908925552283138064184438392780, 2.0965641709531207010864070172, 3.013089882524164033690450806031, 3.65998176623854345186508306874, 4.47860056570431601088114444231, 5.522967906464066301636934934730, 6.53721378297144459837804412911, 7.633857908557369678146463081157, 8.32910721191464872494039076730, 9.02106963081681270719948564624, 9.61428386256655131776710252020, 10.47550766243308917980802888226, 11.05942601078170240339333939951, 11.55093184483020799324995617227, 12.836802549372246967805340316585, 13.28033132234569873264215302143, 14.33067121626578371298668254463, 15.373467585825814422423055105648, 15.71664260779807127668218689883, 16.53859820156254422599146961170, 17.39567148276898188776047945037, 17.77201512308904259492721258892, 18.87939670686393070628459356867, 19.52566485630560726567570262930, 20.11739613128436090779088784855

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