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Properties

Label 1-6025-6025.1711-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.742 + 0.669i$
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  + (0.669 + 0.743i)2-s + (0.669 − 0.743i)3-s + (−0.104 + 0.994i)4-s + 6-s + (0.669 + 0.743i)7-s + (−0.809 + 0.587i)8-s + (−0.104 − 0.994i)9-s + (0.669 + 0.743i)11-s + (0.669 + 0.743i)12-s + (−0.978 + 0.207i)13-s + (−0.104 + 0.994i)14-s + (−0.978 − 0.207i)16-s + (0.309 + 0.951i)17-s + (0.669 − 0.743i)18-s + (−0.104 − 0.994i)19-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)2-s + (0.669 − 0.743i)3-s + (−0.104 + 0.994i)4-s + 6-s + (0.669 + 0.743i)7-s + (−0.809 + 0.587i)8-s + (−0.104 − 0.994i)9-s + (0.669 + 0.743i)11-s + (0.669 + 0.743i)12-s + (−0.978 + 0.207i)13-s + (−0.104 + 0.994i)14-s + (−0.978 − 0.207i)16-s + (0.309 + 0.951i)17-s + (0.669 − 0.743i)18-s + (−0.104 − 0.994i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.742 + 0.669i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1711, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.742 + 0.669i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.036522778 + 2.697352787i\)
\(L(\frac12)\) \(\approx\) \(1.036522778 + 2.697352787i\)
\(L(1)\) \(\approx\) \(1.563419537 + 0.8442381851i\)
\(L(1)\) \(\approx\) \(1.563419537 + 0.8442381851i\)

Euler product

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

−17.45523488617255185734151617838, −16.53450111129352669993609680079, −16.206362475817403510571519067721, −15.196527672783278043095964753703, −14.57651332872483060095202359842, −14.09982629473502115543866460730, −13.87895999696588685473847293929, −12.91638809197047758203119257258, −12.07022493354672587442694343194, −11.55514717391266465363674876221, −10.69231882883151727190596652482, −10.37005690756584999856856045116, −9.53899263364736793125504322362, −9.11615008061940772582757475717, −8.02845026409178554847961027091, −7.60943793044625048875754716336, −6.48133772650516684468786525693, −5.66398535838546479548754531785, −4.91057832613722636852202695691, −4.31292697501155852431795017404, −3.78682770783337878060720155930, −3.0129436937837000475307663423, −2.312621535868260053587052019503, −1.49645982343141031715075363286, −0.47363280900517641455872088719, 1.28665636604621492305577732363, 2.21543442105704246664975908968, 2.588740445730787366150907081712, 3.67564092535350947163481153531, 4.30725518939623547978674113769, 5.08929997287071989703581464853, 5.86692903701830378737140641869, 6.55741010302401772566503006433, 7.22366518369098316052191935420, 7.813596498614865147392687080, 8.349871406752883792021604005061, 9.27273382747333151026756273287, 9.467699122927569883824229169329, 10.93587885904738389157038643480, 11.75785038550948781966036725333, 12.29396247080357475730419582929, 12.675772040533842419205808670548, 13.43217568301875087707695433787, 14.31806596065868736742940883492, 14.63379728824187043040401698017, 15.03479534979106157609019378957, 15.758305849223991981504586015022, 16.66109327612536956065238294645, 17.50610622149328516673084538162, 17.74908866880737033977292758554

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