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Properties

Label 1-6025-6025.866-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.906 + 0.421i$
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  + (0.309 + 0.951i)2-s + 3-s + (−0.809 + 0.587i)4-s + (0.309 + 0.951i)6-s + (−0.309 + 0.951i)7-s + (−0.809 − 0.587i)8-s + 9-s + (−0.309 − 0.951i)11-s + (−0.809 + 0.587i)12-s + (0.809 − 0.587i)13-s − 14-s + (0.309 − 0.951i)16-s + (0.809 − 0.587i)17-s + (0.309 + 0.951i)18-s + (0.809 + 0.587i)19-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + 3-s + (−0.809 + 0.587i)4-s + (0.309 + 0.951i)6-s + (−0.309 + 0.951i)7-s + (−0.809 − 0.587i)8-s + 9-s + (−0.309 − 0.951i)11-s + (−0.809 + 0.587i)12-s + (0.809 − 0.587i)13-s − 14-s + (0.309 − 0.951i)16-s + (0.809 − 0.587i)17-s + (0.309 + 0.951i)18-s + (0.809 + 0.587i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.906 + 0.421i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (866, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.906 + 0.421i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.667909334 + 0.5893583278i\)
\(L(\frac12)\) \(\approx\) \(2.667909334 + 0.5893583278i\)
\(L(1)\) \(\approx\) \(1.455201638 + 0.6463931971i\)
\(L(1)\) \(\approx\) \(1.455201638 + 0.6463931971i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T \)
3 \( 1 + T \)
7 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−18.08600251581168527812453862916, −17.049332095019054253236979928542, −16.2916271167791270051751083845, −15.511648362402132076656602960324, −14.78966070933497203167518875143, −14.232073610418037901216062395751, −13.65183264932628625109765981699, −13.12880321194979103650640182547, −12.54721563792107980357554258581, −11.84649922215196705693432514518, −10.874753714106112738032873769790, −10.36911017585035572291768529290, −9.69027425477220552543964391784, −9.32218093428475954246434643805, −8.29824429052379528208448200458, −7.86251158513693665938606752504, −6.83264222852028422888772259707, −6.31573141938350509609868038915, −4.90388462313937841160933425882, −4.618837570514546215676444280269, −3.62727288166222277865904872996, −3.33372550845466334153692616046, −2.45608512718714280749178948294, −1.52092121895536938917148283818, −1.11506932743953292771991030705, 0.56030617308018690367152634256, 1.77574175663184076316254820771, 2.828691649990582209424346728515, 3.46998848440850583486790231017, 3.74640663983522437536932797469, 5.082498914868091701624569449282, 5.60912319000974654621905239425, 6.122291816553853123064806973240, 7.13770151143311551764477876594, 7.78528237516476483867965313372, 8.346266478084629398492917106526, 8.80804267366201693107006252254, 9.66884139329924833838624605078, 10.00648538477665807482531595641, 11.325193731570821625139017224641, 12.047907248035196551688914253881, 12.73928729493892082642769884316, 13.55406285472241523666955086808, 13.71352691933526962591611348270, 14.532951993516510593035180744028, 15.23764502494078348962324995151, 15.76187952980412616613255345999, 16.09935074085400158296641735609, 16.86491835041347029788546813577, 17.92560622289282597258833676377

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