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Properties

Label 1-6025-6025.5206-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.334 - 0.942i$
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  + (−0.913 − 0.406i)2-s + (0.104 + 0.994i)3-s + (0.669 + 0.743i)4-s + (0.309 − 0.951i)6-s + (0.207 + 0.978i)7-s + (−0.309 − 0.951i)8-s + (−0.978 + 0.207i)9-s + (0.406 − 0.913i)11-s + (−0.669 + 0.743i)12-s + (0.207 − 0.978i)13-s + (0.207 − 0.978i)14-s + (−0.104 + 0.994i)16-s + i·17-s + (0.978 + 0.207i)18-s + (−0.743 − 0.669i)19-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)2-s + (0.104 + 0.994i)3-s + (0.669 + 0.743i)4-s + (0.309 − 0.951i)6-s + (0.207 + 0.978i)7-s + (−0.309 − 0.951i)8-s + (−0.978 + 0.207i)9-s + (0.406 − 0.913i)11-s + (−0.669 + 0.743i)12-s + (0.207 − 0.978i)13-s + (0.207 − 0.978i)14-s + (−0.104 + 0.994i)16-s + i·17-s + (0.978 + 0.207i)18-s + (−0.743 − 0.669i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.334 - 0.942i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (5206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.334 - 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5088321980 - 0.3594708116i\)
\(L(\frac12)\) \(\approx\) \(0.5088321980 - 0.3594708116i\)
\(L(1)\) \(\approx\) \(0.6701419640 + 0.08343912790i\)
\(L(1)\) \(\approx\) \(0.6701419640 + 0.08343912790i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.913 - 0.406i)T \)
3 \( 1 + (0.104 + 0.994i)T \)
7 \( 1 + (0.207 + 0.978i)T \)
11 \( 1 + (0.406 - 0.913i)T \)
13 \( 1 + (0.207 - 0.978i)T \)
17 \( 1 + iT \)
19 \( 1 + (-0.743 - 0.669i)T \)
23 \( 1 - iT \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 - T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 - T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 + (-0.951 - 0.309i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (-0.104 + 0.994i)T \)
89 \( 1 + (-0.406 + 0.913i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−17.847265539801574881243494199361, −17.126915093644466477695720898905, −16.9348584720379827697281952786, −16.01146555014588062525674101852, −15.30365222607359148257462514968, −14.33231144191567493648628017921, −14.1346034279949950034011502189, −13.422437147101857467081649883290, −12.47282647635107847049079863958, −11.66999633110702444130379720005, −11.40461791661734701894160168121, −10.4078668546508532868584651606, −9.75242213443075014490382063917, −9.13493714792957717203661928794, −8.305777260155031990722788366674, −7.754663815213594370968038522107, −7.03638906121727607195767257466, −6.74197845984409197582010026033, −6.025483617308934799263538310323, −5.016310432977581781554443740530, −4.24023314619809647231551583289, −3.178342657267363539316256799512, −2.13260292314051309343476977312, −1.61448781077611818553326244002, −0.92971651287860420436630413890, 0.23762190361994427831883011339, 1.3731881468045307296150447946, 2.43017939694865909039080917615, 2.952643411583626534486763309445, 3.63156474425085536645056375402, 4.4745483894013649367880886373, 5.38552594958940560103158186045, 6.1367555289164019425241462006, 6.69218343717520024842271517394, 8.09304371350459762406902820624, 8.467451038249198329936847799, 8.75843004847307850869869998558, 9.60173057801679422323184340858, 10.38734195178660043120303727656, 10.769975360144049445210103633751, 11.41933367999637709909074701791, 12.1028278325470860160950400312, 12.78572680728782297319314817604, 13.585762395246383221629883190763, 14.70018928074927638045967459265, 15.07361845025557034659079670381, 15.71788408212063763940122905775, 16.34507495597059801764359304953, 16.918439878932158290590092863945, 17.63449767560924471843037048941

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