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Properties

Label 1-6025-6025.41-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.0676 - 0.997i$
Analytic cond. $27.9799$
Root an. cond. $27.9799$
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  + (0.951 − 0.309i)2-s + (0.951 − 0.309i)3-s + (0.809 − 0.587i)4-s + (0.809 − 0.587i)6-s + (0.987 + 0.156i)7-s + (0.587 − 0.809i)8-s + (0.809 − 0.587i)9-s + (0.453 − 0.891i)11-s + (0.587 − 0.809i)12-s + (0.987 − 0.156i)13-s + (0.987 − 0.156i)14-s + (0.309 − 0.951i)16-s + (−0.707 + 0.707i)17-s + (0.587 − 0.809i)18-s + (−0.156 − 0.987i)19-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.951 − 0.309i)3-s + (0.809 − 0.587i)4-s + (0.809 − 0.587i)6-s + (0.987 + 0.156i)7-s + (0.587 − 0.809i)8-s + (0.809 − 0.587i)9-s + (0.453 − 0.891i)11-s + (0.587 − 0.809i)12-s + (0.987 − 0.156i)13-s + (0.987 − 0.156i)14-s + (0.309 − 0.951i)16-s + (−0.707 + 0.707i)17-s + (0.587 − 0.809i)18-s + (−0.156 − 0.987i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0676 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0676 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.0676 - 0.997i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.0676 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.447393688 - 4.759222460i\)
\(L(\frac12)\) \(\approx\) \(4.447393688 - 4.759222460i\)
\(L(1)\) \(\approx\) \(2.735488422 - 1.369699321i\)
\(L(1)\) \(\approx\) \(2.735488422 - 1.369699321i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.951 - 0.309i)T \)
3 \( 1 + (0.951 - 0.309i)T \)
7 \( 1 + (0.987 + 0.156i)T \)
11 \( 1 + (0.453 - 0.891i)T \)
13 \( 1 + (0.987 - 0.156i)T \)
17 \( 1 + (-0.707 + 0.707i)T \)
19 \( 1 + (-0.156 - 0.987i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + iT \)
31 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (0.951 - 0.309i)T \)
43 \( 1 + (-0.987 + 0.156i)T \)
47 \( 1 + iT \)
53 \( 1 - iT \)
59 \( 1 - iT \)
61 \( 1 + iT \)
67 \( 1 - iT \)
71 \( 1 + (-0.707 + 0.707i)T \)
73 \( 1 + (0.987 + 0.156i)T \)
79 \( 1 + (-0.587 + 0.809i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (-0.891 - 0.453i)T \)
97 \( 1 - T \)
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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

−17.89189853556396864724920255894, −17.00224841107443043732929959879, −16.42772038446415180458863142824, −15.493524967389735862627115561221, −15.30506930406062234786881916536, −14.52733423797092147102869336575, −14.001279417743285706140023292481, −13.5219319303671837821303843197, −12.894248311824565557840467717434, −11.88316330884218894607133655062, −11.5345722280333470524630371381, −10.64837365134111521775507576292, −9.96471005840005962155600713807, −9.060042980951242898568805524231, −8.37810511196391229963812949036, −7.730972734701473744632857888923, −7.25156596355397450412053140478, −6.38067016125525161555264919558, −5.55460984039891000611384099929, −4.72398093248826696831007918768, −4.11078619328865496697574971937, −3.76781647955198515979861360504, −2.743267694971223464441309864877, −1.83906344010474049808778029079, −1.59029213965924117254030327371, 0.92947958333121651897915974308, 1.6437395145184600107948909420, 2.24613706271751494509269996625, 3.119565067583640323084325310930, 3.773690877826578983525965006800, 4.34497130477235507016884337937, 5.189661579562454827756571364531, 6.03762939958209077340970652835, 6.66938028237675607961772291834, 7.33856564639340918653102999439, 8.43992076046771281875495186723, 8.57001268468745236534183794958, 9.46620046521522603898629870267, 10.63974680145789068354927579699, 10.94148445958379891825598182928, 11.61542811601401357264973642203, 12.55389743910520164947733624866, 12.949371453615406440919816620743, 13.73661677525638131697901700618, 14.261609669996535171614917365121, 14.585095516791054042941734241322, 15.4855662794383865412554115741, 15.85272324974938953242653913864, 16.697930213108724723022590654935, 17.82900947714687515805186080172

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