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Properties

Label 1-6025-6025.3272-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.721 - 0.692i$
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  + (0.156 + 0.987i)2-s + (0.156 + 0.987i)3-s + (−0.951 + 0.309i)4-s + (−0.951 + 0.309i)6-s + (0.0784 − 0.996i)7-s + (−0.453 − 0.891i)8-s + (−0.951 + 0.309i)9-s + (−0.852 + 0.522i)11-s + (−0.453 − 0.891i)12-s + (0.0784 + 0.996i)13-s + (0.996 − 0.0784i)14-s + (0.809 − 0.587i)16-s + (0.923 − 0.382i)17-s + (−0.453 − 0.891i)18-s + (0.649 − 0.760i)19-s + ⋯
L(s)  = 1  + (0.156 + 0.987i)2-s + (0.156 + 0.987i)3-s + (−0.951 + 0.309i)4-s + (−0.951 + 0.309i)6-s + (0.0784 − 0.996i)7-s + (−0.453 − 0.891i)8-s + (−0.951 + 0.309i)9-s + (−0.852 + 0.522i)11-s + (−0.453 − 0.891i)12-s + (0.0784 + 0.996i)13-s + (0.996 − 0.0784i)14-s + (0.809 − 0.587i)16-s + (0.923 − 0.382i)17-s + (−0.453 − 0.891i)18-s + (0.649 − 0.760i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.721 - 0.692i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (3272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.721 - 0.692i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3269217034 + 0.8133507389i\)
\(L(\frac12)\) \(\approx\) \(-0.3269217034 + 0.8133507389i\)
\(L(1)\) \(\approx\) \(0.6326891965 + 0.6671648593i\)
\(L(1)\) \(\approx\) \(0.6326891965 + 0.6671648593i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.156 + 0.987i)T \)
3 \( 1 + (0.156 + 0.987i)T \)
7 \( 1 + (0.0784 - 0.996i)T \)
11 \( 1 + (-0.852 + 0.522i)T \)
13 \( 1 + (0.0784 + 0.996i)T \)
17 \( 1 + (0.923 - 0.382i)T \)
19 \( 1 + (0.649 - 0.760i)T \)
23 \( 1 + (-0.923 - 0.382i)T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + (0.382 + 0.923i)T \)
37 \( 1 + (0.923 + 0.382i)T \)
41 \( 1 + (-0.987 + 0.156i)T \)
43 \( 1 + (0.996 - 0.0784i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (0.707 + 0.707i)T \)
67 \( 1 + (0.707 + 0.707i)T \)
71 \( 1 + (-0.382 - 0.923i)T \)
73 \( 1 + (0.0784 - 0.996i)T \)
79 \( 1 + (-0.453 - 0.891i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (-0.233 + 0.972i)T \)
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

−17.61743372903056224282565262043, −16.92200024787530503464124859, −15.854473007225185386791318504698, −15.20484988617551991596168808898, −14.31870752513689553038251295098, −14.02224551561151393419627172014, −13.036094509767267413947252600878, −12.67854218804120290825076387622, −12.20101429599256971268844160087, −11.43156131617858819052583874872, −10.92077434221296645473616074607, −9.97466354808883023723225388821, −9.4420759816806105551551867001, −8.478229500065135115997304578987, −8.02676396538962468968744722089, −7.57895720847964548541701755784, −6.10850534294986346917188240045, −5.613259670101456030745502490384, −5.381390981777497870300288343030, −4.02147362308080390924144578686, −3.11051616323597511322921043205, −2.81407434853368409094629932540, −1.914344308656668901166193006658, −1.258524884642413125226855524, −0.24302450564167884302114593165, 0.93535492273879095297974158870, 2.33749104122154243536878851367, 3.27579208518751842501554086008, 3.95039875185600587458248852650, 4.60582320693045855538279883240, 5.055595370490196402235343200044, 5.86407189579844840191220150337, 6.70831954476526435359301436969, 7.48516076369979272722697071249, 7.913750488530608122099409639866, 8.75814945342144685253049806107, 9.52328687438662947944733925981, 9.98487338309072148295820785132, 10.57107025258770913478540207702, 11.5466870304655663569377969000, 12.17503754814622132330005737480, 13.31172399890130798439658133516, 13.674081599117465110304384238244, 14.343506851194601235141515376830, 14.81469408140027287696283737364, 15.69580980762135310263090514145, 16.07311696026799470125061506188, 16.66833827225480918494044468964, 17.18762971297992173764271413310, 17.90962766504823873206717365383

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