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Properties

Label 1-6025-6025.1142-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.0679 - 0.997i$
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  + (−0.777 − 0.629i)2-s + (−0.544 + 0.838i)3-s + (0.207 + 0.978i)4-s + (0.951 − 0.309i)6-s + (−0.130 + 0.991i)7-s + (0.453 − 0.891i)8-s + (−0.406 − 0.913i)9-s + (0.878 − 0.477i)11-s + (−0.933 − 0.358i)12-s + (−0.477 + 0.878i)13-s + (0.725 − 0.688i)14-s + (−0.913 + 0.406i)16-s + (0.0784 − 0.996i)17-s + (−0.258 + 0.965i)18-s + (−0.983 + 0.182i)19-s + ⋯
L(s)  = 1  + (−0.777 − 0.629i)2-s + (−0.544 + 0.838i)3-s + (0.207 + 0.978i)4-s + (0.951 − 0.309i)6-s + (−0.130 + 0.991i)7-s + (0.453 − 0.891i)8-s + (−0.406 − 0.913i)9-s + (0.878 − 0.477i)11-s + (−0.933 − 0.358i)12-s + (−0.477 + 0.878i)13-s + (0.725 − 0.688i)14-s + (−0.913 + 0.406i)16-s + (0.0784 − 0.996i)17-s + (−0.258 + 0.965i)18-s + (−0.983 + 0.182i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0679 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0679 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.0679 - 0.997i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.0679 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03247733557 - 0.03476624322i\)
\(L(\frac12)\) \(\approx\) \(0.03247733557 - 0.03476624322i\)
\(L(1)\) \(\approx\) \(0.4822178338 + 0.1058350408i\)
\(L(1)\) \(\approx\) \(0.4822178338 + 0.1058350408i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.777 - 0.629i)T \)
3 \( 1 + (-0.544 + 0.838i)T \)
7 \( 1 + (-0.130 + 0.991i)T \)
11 \( 1 + (0.878 - 0.477i)T \)
13 \( 1 + (-0.477 + 0.878i)T \)
17 \( 1 + (0.0784 - 0.996i)T \)
19 \( 1 + (-0.983 + 0.182i)T \)
23 \( 1 + (-0.522 + 0.852i)T \)
29 \( 1 + (-0.998 - 0.0523i)T \)
31 \( 1 + (-0.566 + 0.824i)T \)
37 \( 1 + (-0.688 + 0.725i)T \)
41 \( 1 + (-0.156 - 0.987i)T \)
43 \( 1 + (-0.382 + 0.923i)T \)
47 \( 1 + (-0.891 + 0.453i)T \)
53 \( 1 + (-0.544 + 0.838i)T \)
59 \( 1 + (-0.933 - 0.358i)T \)
61 \( 1 + (-0.156 + 0.987i)T \)
67 \( 1 + (0.838 - 0.544i)T \)
71 \( 1 + (-0.942 + 0.333i)T \)
73 \( 1 + (-0.522 + 0.852i)T \)
79 \( 1 + (0.891 - 0.453i)T \)
83 \( 1 + (0.978 + 0.207i)T \)
89 \( 1 + (0.958 + 0.284i)T \)
97 \( 1 + (0.978 - 0.207i)T \)
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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

−17.66019217972768717630761489349, −17.23471704546280188167459492763, −16.80483140861640533199360250525, −16.30813196331566619885111533920, −15.18108532642066910542389189128, −14.70006301209063343470451040375, −14.11865511880009959407457819658, −13.1497799688734784893808068451, −12.777185186366998846276972702918, −11.89117317467499347803712719338, −11.0776944625877855108274998490, −10.52285793671524892456825038823, −10.04093132278814873322535249008, −9.13169086144419929478393513215, −8.27889522068784676832830247014, −7.75181278951011133812311274462, −7.157950408714223179198262350136, −6.45627041774662145200076527736, −6.100178394881857918851396517753, −5.13972441308464498176873792206, −4.42480723903589877209625820083, −3.47949216271579624553739001301, −2.03719893383572097268203837604, −1.76107602439710329182396284462, −0.649203228431266165645132319789, 0.02488841732398362467752364456, 1.38923236499765165595520082125, 2.098944398442371133576010000647, 3.105226133616637269332874757770, 3.62602517987407422828602953611, 4.47023830864511768495855470044, 5.18340122193152129534259565390, 6.14926269958153203088130673755, 6.65936223161283404180549494655, 7.566996788009146672848590853523, 8.58263844051755496487987448806, 9.14026092970136407143805610625, 9.432634750387490551246763241488, 10.16055385576073287805106598146, 11.033177109642372271767613786249, 11.519433063454418207246759840226, 12.02238098910269126753557053446, 12.49309196925200620175394423855, 13.54364184358613822130030641192, 14.40646945822727253565042864901, 15.03998222820054144912971725544, 15.954063456307611654126908045647, 16.22703348036079166122104249295, 17.01946059154136284208220311155, 17.40777103863046144623910753183

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