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Properties

Label 1-6025-6025.3047-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.512 + 0.858i$
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  + (−0.629 − 0.777i)2-s + (0.777 + 0.629i)3-s + (−0.207 + 0.978i)4-s − i·6-s + (−0.958 + 0.284i)7-s + (0.891 − 0.453i)8-s + (0.207 + 0.978i)9-s + (−0.284 − 0.958i)11-s + (−0.777 + 0.629i)12-s + (−0.0261 − 0.999i)13-s + (0.824 + 0.566i)14-s + (−0.913 − 0.406i)16-s + (−0.233 − 0.972i)17-s + (0.629 − 0.777i)18-s + (−0.566 − 0.824i)19-s + ⋯
L(s)  = 1  + (−0.629 − 0.777i)2-s + (0.777 + 0.629i)3-s + (−0.207 + 0.978i)4-s − i·6-s + (−0.958 + 0.284i)7-s + (0.891 − 0.453i)8-s + (0.207 + 0.978i)9-s + (−0.284 − 0.958i)11-s + (−0.777 + 0.629i)12-s + (−0.0261 − 0.999i)13-s + (0.824 + 0.566i)14-s + (−0.913 − 0.406i)16-s + (−0.233 − 0.972i)17-s + (0.629 − 0.777i)18-s + (−0.566 − 0.824i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.512 + 0.858i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (3047, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.512 + 0.858i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07193150966 + 0.1267192439i\)
\(L(\frac12)\) \(\approx\) \(0.07193150966 + 0.1267192439i\)
\(L(1)\) \(\approx\) \(0.7166494853 - 0.1433804693i\)
\(L(1)\) \(\approx\) \(0.7166494853 - 0.1433804693i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.629 - 0.777i)T \)
3 \( 1 + (0.777 + 0.629i)T \)
7 \( 1 + (-0.958 + 0.284i)T \)
11 \( 1 + (-0.284 - 0.958i)T \)
13 \( 1 + (-0.0261 - 0.999i)T \)
17 \( 1 + (-0.233 - 0.972i)T \)
19 \( 1 + (-0.566 - 0.824i)T \)
23 \( 1 + (0.996 - 0.0784i)T \)
29 \( 1 + (0.629 + 0.777i)T \)
31 \( 1 + (-0.688 - 0.725i)T \)
37 \( 1 + (-0.942 - 0.333i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (-0.972 + 0.233i)T \)
47 \( 1 + (-0.156 - 0.987i)T \)
53 \( 1 + (-0.358 + 0.933i)T \)
59 \( 1 + (0.998 - 0.0523i)T \)
61 \( 1 + (0.453 + 0.891i)T \)
67 \( 1 + (0.933 - 0.358i)T \)
71 \( 1 + (-0.477 + 0.878i)T \)
73 \( 1 + (-0.852 - 0.522i)T \)
79 \( 1 + (-0.707 - 0.707i)T \)
83 \( 1 + (0.104 - 0.994i)T \)
89 \( 1 + (-0.477 + 0.878i)T \)
97 \( 1 + (-0.913 + 0.406i)T \)
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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

−17.41742127654010560186693537966, −16.95325413572505900459956742378, −16.18106798649248482411619474406, −15.47447401118310177296244818587, −14.92776325087466911620464975583, −14.34534847823072529264564816693, −13.6752297332036846204428935398, −12.94056124341434807569351854350, −12.54858358572874647444175164270, −11.5524279069028612865932359567, −10.54058353248484155463117584871, −9.86961714897027216424372884750, −9.508830362708347098062844554447, −8.56541000439567480181010734740, −8.26489572955281691690077237364, −7.28939407982286859037367820705, −6.69815977551547204657905703996, −6.52906663341550766241378097601, −5.46938623189757739035018729091, −4.455664088275597820856858518715, −3.812457019741186187581746953481, −2.83275332977586881021206536431, −1.81555857572482681844047278624, −1.46158637013547962690776293333, −0.04622647278917413953507080311, 0.879730415880520220991760252863, 2.1711330848458062720214431332, 2.8353481283341969845012617248, 3.21025566482492059004173015314, 3.86208123129315915154603547502, 4.92037205524732300371087006708, 5.47481538768088088041395576964, 6.778223039756969836308759482864, 7.30562133681899144876736499088, 8.26593052607738144142496117905, 8.84807756463644433037837117406, 9.12678339718503035160624523784, 10.05905614165615426382776836168, 10.47847268493393538275130242599, 11.10512007224283212094813994219, 11.86889852259786671526652239519, 12.85695683376851055353672202736, 13.236168934884747054390213837293, 13.71487117125784973093189814986, 14.73489947746162149739052993491, 15.5290961863848429395200721204, 15.98891471049807150640921398170, 16.56760818715056902886200673030, 17.26269592544209875464612259570, 18.13913886003799577423257545943

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