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Properties

Label 1-6025-6025.3989-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.117 - 0.993i$
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  + (−0.743 + 0.669i)2-s + (−0.406 + 0.913i)3-s + (0.104 − 0.994i)4-s + (−0.309 − 0.951i)6-s + (0.629 − 0.777i)7-s + (0.587 + 0.809i)8-s + (−0.669 − 0.743i)9-s + (0.0523 − 0.998i)11-s + (0.866 + 0.5i)12-s + (−0.965 − 0.258i)13-s + (0.0523 + 0.998i)14-s + (−0.978 − 0.207i)16-s + (−0.453 − 0.891i)17-s + (0.994 + 0.104i)18-s + (0.629 − 0.777i)19-s + ⋯
L(s)  = 1  + (−0.743 + 0.669i)2-s + (−0.406 + 0.913i)3-s + (0.104 − 0.994i)4-s + (−0.309 − 0.951i)6-s + (0.629 − 0.777i)7-s + (0.587 + 0.809i)8-s + (−0.669 − 0.743i)9-s + (0.0523 − 0.998i)11-s + (0.866 + 0.5i)12-s + (−0.965 − 0.258i)13-s + (0.0523 + 0.998i)14-s + (−0.978 − 0.207i)16-s + (−0.453 − 0.891i)17-s + (0.994 + 0.104i)18-s + (0.629 − 0.777i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.117 - 0.993i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (3989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.117 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6380014159 - 0.5670619996i\)
\(L(\frac12)\) \(\approx\) \(0.6380014159 - 0.5670619996i\)
\(L(1)\) \(\approx\) \(0.6541560642 + 0.05110285229i\)
\(L(1)\) \(\approx\) \(0.6541560642 + 0.05110285229i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.743 + 0.669i)T \)
3 \( 1 + (-0.406 + 0.913i)T \)
7 \( 1 + (0.629 - 0.777i)T \)
11 \( 1 + (0.0523 - 0.998i)T \)
13 \( 1 + (-0.965 - 0.258i)T \)
17 \( 1 + (-0.453 - 0.891i)T \)
19 \( 1 + (0.629 - 0.777i)T \)
23 \( 1 + (-0.156 - 0.987i)T \)
29 \( 1 + (0.207 - 0.978i)T \)
31 \( 1 + (-0.998 - 0.0523i)T \)
37 \( 1 + (0.629 - 0.777i)T \)
41 \( 1 + (-0.587 + 0.809i)T \)
43 \( 1 + (0.987 - 0.156i)T \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (0.743 + 0.669i)T \)
59 \( 1 + (-0.994 + 0.104i)T \)
61 \( 1 + (0.587 - 0.809i)T \)
67 \( 1 + (-0.743 + 0.669i)T \)
71 \( 1 + (0.838 - 0.544i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + (-0.951 - 0.309i)T \)
83 \( 1 + (0.913 - 0.406i)T \)
89 \( 1 + (-0.998 - 0.0523i)T \)
97 \( 1 + (0.669 - 0.743i)T \)
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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

−17.92542277956123753112974453996, −17.41062963853757875930071478026, −16.94058864016404691106951338499, −16.12456688930394215537086955747, −15.24733547297677421165179585669, −14.56324858184877007772248426757, −13.78421361490296256355363937114, −12.91246410633760953669122250302, −12.242359823563299891741534445, −12.14035458643850152130360071756, −11.31876589410516769634112545348, −10.6994489334787740978514747678, −9.91029826546047935947242768307, −9.178388658775244075893762461819, −8.54808053815654470043471978749, −7.68711413553950754521736831354, −7.41478037568573261091676916425, −6.61400433753117481516442743250, −5.65577979102264196405710660164, −4.98666762527124395381576923947, −4.11974752335937480280929285804, −3.09334979533127966944574248131, −2.12604062363109391251465028628, −1.88663892589186670866676952131, −1.10636879846937381678402464566, 0.4265736514760198520306377954, 0.77137044019769299564476591573, 2.21869011338905879764655899602, 3.00080755564666082528466290883, 4.18781900408549638317556607260, 4.68229523426469473636898898661, 5.382560867265877551526130434037, 5.99117313514979261715948343907, 6.89802268239673674060016502258, 7.471605353654352449928211100303, 8.22646322815371418104332658607, 9.00572607936238782914926149697, 9.4865815851307274840464277275, 10.251269658367506789189855287587, 10.819708301782623191619862644743, 11.32940763839738375905315712232, 11.91275251302474235520435986437, 13.15696767529967055723823575614, 14.13281018939305967144492646842, 14.238919525598137346250369417873, 15.177868855169027426683979026524, 15.691789787490304631674130996527, 16.412574750233877322880622892471, 16.854008566140424156241922268216, 17.36141038998147303663555538912

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