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Properties

Label 1-6025-6025.1673-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.795 - 0.606i$
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  + (0.0523 − 0.998i)2-s + (−0.358 − 0.933i)3-s + (−0.994 − 0.104i)4-s + (−0.951 + 0.309i)6-s + (0.958 + 0.284i)7-s + (−0.156 + 0.987i)8-s + (−0.743 + 0.669i)9-s + (0.333 + 0.942i)11-s + (0.258 + 0.965i)12-s + (−0.608 + 0.793i)13-s + (0.333 − 0.942i)14-s + (0.978 + 0.207i)16-s + (0.996 − 0.0784i)17-s + (0.629 + 0.777i)18-s + (0.958 + 0.284i)19-s + ⋯
L(s)  = 1  + (0.0523 − 0.998i)2-s + (−0.358 − 0.933i)3-s + (−0.994 − 0.104i)4-s + (−0.951 + 0.309i)6-s + (0.958 + 0.284i)7-s + (−0.156 + 0.987i)8-s + (−0.743 + 0.669i)9-s + (0.333 + 0.942i)11-s + (0.258 + 0.965i)12-s + (−0.608 + 0.793i)13-s + (0.333 − 0.942i)14-s + (0.978 + 0.207i)16-s + (0.996 − 0.0784i)17-s + (0.629 + 0.777i)18-s + (0.958 + 0.284i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.795 - 0.606i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.795 - 0.606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.580900096 - 0.5340218735i\)
\(L(\frac12)\) \(\approx\) \(1.580900096 - 0.5340218735i\)
\(L(1)\) \(\approx\) \(0.8993377179 - 0.5208858722i\)
\(L(1)\) \(\approx\) \(0.8993377179 - 0.5208858722i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.0523 - 0.998i)T \)
3 \( 1 + (-0.358 - 0.933i)T \)
7 \( 1 + (0.958 + 0.284i)T \)
11 \( 1 + (0.333 + 0.942i)T \)
13 \( 1 + (-0.608 + 0.793i)T \)
17 \( 1 + (0.996 - 0.0784i)T \)
19 \( 1 + (0.958 + 0.284i)T \)
23 \( 1 + (0.852 - 0.522i)T \)
29 \( 1 + (-0.544 - 0.838i)T \)
31 \( 1 + (0.902 + 0.430i)T \)
37 \( 1 + (-0.958 - 0.284i)T \)
41 \( 1 + (0.987 - 0.156i)T \)
43 \( 1 + (0.972 + 0.233i)T \)
47 \( 1 + (0.891 + 0.453i)T \)
53 \( 1 + (-0.998 + 0.0523i)T \)
59 \( 1 + (-0.777 - 0.629i)T \)
61 \( 1 + (0.156 + 0.987i)T \)
67 \( 1 + (-0.0523 + 0.998i)T \)
71 \( 1 + (0.983 - 0.182i)T \)
73 \( 1 + (-0.382 + 0.923i)T \)
79 \( 1 + (0.453 + 0.891i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (-0.902 - 0.430i)T \)
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

−17.468991631947505379549504082285, −17.008847250918254042210716506064, −16.594910958823355675086598378993, −15.664219349808825137121268888495, −15.367494556189439172342224954126, −14.48680220170120642867992357871, −14.19235667200072810371889139576, −13.50749273301085963669469556705, −12.459885535672633334268321031855, −11.86149114225616785143369573796, −10.99365983309104365722268010604, −10.51103981504350134892732815964, −9.605873590780946521510687824828, −9.14510395564423552303736683200, −8.334229628027831474463868041069, −7.72033501894461680772761221667, −7.09886365735508037340069651313, −6.04291548707810812372713388135, −5.45556854176283535829373051574, −5.067043655669228548195000153675, −4.326925081059145898360443706429, −3.44235126447175731372121071964, −3.00122955956611846095414787976, −1.25796660501174325360256346074, −0.55348475044397849847088594818, 1.022124210173141946752247668526, 1.41106792583120366987026002913, 2.31217683024635813076304348627, 2.74333948806686732303189466780, 3.96557940961079308203738460626, 4.69940932110058661426921697558, 5.276027863333927791593662821175, 5.94840951046350137041179423197, 7.06606264049712838623040861397, 7.59273830358888977639435807760, 8.2550632427561310824511886119, 9.121381076695242380507110064896, 9.67860520872860952021843673390, 10.54023635375933596544438951720, 11.28216054210713912127270616286, 11.76890906125291472227631943322, 12.38294490104305450662820657507, 12.614650972948211401102981468478, 13.75054186767342954336296824573, 14.26725342581762909843556196682, 14.54709531924167319089375050747, 15.59208207557759919689398193084, 16.73147226302600412507991870269, 17.35412209874861650207398749502, 17.621007009191658218702783384093

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