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Properties

Label 1-2675-2675.184-r1-0-0
Degree $1$
Conductor $2675$
Sign $-0.970 + 0.241i$
Analytic cond. $287.468$
Root an. cond. $287.468$
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  + (−0.386 + 0.922i)2-s + (0.451 − 0.892i)3-s + (−0.700 − 0.713i)4-s + (0.648 + 0.761i)6-s + (0.674 − 0.737i)7-s + (0.928 − 0.370i)8-s + (−0.592 − 0.805i)9-s + (−0.364 − 0.931i)11-s + (−0.952 + 0.303i)12-s + (0.749 − 0.661i)13-s + (0.419 + 0.907i)14-s + (−0.0177 + 0.999i)16-s + (0.741 − 0.670i)17-s + (0.972 − 0.234i)18-s + (−0.905 + 0.424i)19-s + ⋯
L(s)  = 1  + (−0.386 + 0.922i)2-s + (0.451 − 0.892i)3-s + (−0.700 − 0.713i)4-s + (0.648 + 0.761i)6-s + (0.674 − 0.737i)7-s + (0.928 − 0.370i)8-s + (−0.592 − 0.805i)9-s + (−0.364 − 0.931i)11-s + (−0.952 + 0.303i)12-s + (0.749 − 0.661i)13-s + (0.419 + 0.907i)14-s + (−0.0177 + 0.999i)16-s + (0.741 − 0.670i)17-s + (0.972 − 0.234i)18-s + (−0.905 + 0.424i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2675\)    =    \(5^{2} \cdot 107\)
Sign: $-0.970 + 0.241i$
Analytic conductor: \(287.468\)
Root analytic conductor: \(287.468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2675} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2675,\ (1:\ ),\ -0.970 + 0.241i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1267586328 - 1.036333874i\)
\(L(\frac12)\) \(\approx\) \(-0.1267586328 - 1.036333874i\)
\(L(1)\) \(\approx\) \(0.8986605269 - 0.2543100865i\)
\(L(1)\) \(\approx\) \(0.8986605269 - 0.2543100865i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
107 \( 1 \)
good2 \( 1 + (-0.386 + 0.922i)T \)
3 \( 1 + (0.451 - 0.892i)T \)
7 \( 1 + (0.674 - 0.737i)T \)
11 \( 1 + (-0.364 - 0.931i)T \)
13 \( 1 + (0.749 - 0.661i)T \)
17 \( 1 + (0.741 - 0.670i)T \)
19 \( 1 + (-0.905 + 0.424i)T \)
23 \( 1 + (-0.194 - 0.980i)T \)
29 \( 1 + (0.989 + 0.141i)T \)
31 \( 1 + (0.553 + 0.832i)T \)
37 \( 1 + (-0.801 - 0.597i)T \)
41 \( 1 + (-0.472 + 0.881i)T \)
43 \( 1 + (-0.430 - 0.902i)T \)
47 \( 1 + (-0.692 + 0.721i)T \)
53 \( 1 + (0.229 - 0.973i)T \)
59 \( 1 + (-0.440 - 0.897i)T \)
61 \( 1 + (-0.493 + 0.869i)T \)
67 \( 1 + (-0.0651 + 0.997i)T \)
71 \( 1 + (0.988 - 0.153i)T \)
73 \( 1 + (-0.822 - 0.568i)T \)
79 \( 1 + (-0.999 + 0.0355i)T \)
83 \( 1 + (-0.657 + 0.753i)T \)
89 \( 1 + (0.523 + 0.851i)T \)
97 \( 1 + (0.503 - 0.864i)T \)
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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

−19.59074383575119190494704848893, −18.88752420355737952213746266671, −18.29805973676398464961622301689, −17.30503810472960157622952687526, −17.01586376435196480043659301734, −15.82809321017114522033690088910, −15.327669250233032837602150835945, −14.54500780471918066299011127000, −13.789837339810792327368978559474, −13.08172978599470960688293642799, −12.08708912615946276236172250291, −11.60150200702083295376219291265, −10.76177693498410241974869558160, −10.16645600396448180945272829587, −9.489457877221224706263769165474, −8.70341974653171375860445021517, −8.2669380285310198045747873102, −7.494388021087699063467369030455, −6.13159014211118356952065509877, −5.07450950112657176996755021087, −4.52389049812575363824285495350, −3.77468617232335261041870649173, −2.89477814682970743772621345616, −2.059020129489308152530467717557, −1.47121976977387724552214624962, 0.20123421630446566022153838162, 0.91053873694397177170304444197, 1.61591146615114570655796179274, 2.91419352136096881704424809837, 3.77728657689558192627058715026, 4.84303242276521503614474555222, 5.66287103297869422050871567518, 6.45502538054852168533839942940, 7.03522447405589394878833733175, 8.06756830968779158510562129396, 8.22793198993108889730627714912, 8.86415403390712959242334825815, 10.14623939014295493703135868393, 10.6016773335551753372842410386, 11.54463136009188311191002828222, 12.58774693809009369866267567708, 13.294757546345954256696243195608, 14.0391460964980928857493073606, 14.27638934325352270099714858161, 15.14161085064062785424682143035, 16.05864350726125808111468195381, 16.63934994434554425567303260306, 17.48121893599556108624106619293, 18.01994358952000624610804016115, 18.619674992804209271058105807228

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