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Properties

Label 1-6025-6025.2148-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.981 + 0.191i$
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  + (−0.891 + 0.453i)2-s + (−0.707 − 0.707i)3-s + (0.587 − 0.809i)4-s + (0.951 + 0.309i)6-s + (0.996 + 0.0784i)7-s + (−0.156 + 0.987i)8-s + i·9-s + (−0.649 + 0.760i)11-s + (−0.987 + 0.156i)12-s + (0.233 − 0.972i)13-s + (−0.923 + 0.382i)14-s + (−0.309 − 0.951i)16-s + (−0.233 + 0.972i)17-s + (−0.453 − 0.891i)18-s + (0.233 + 0.972i)19-s + ⋯
L(s)  = 1  + (−0.891 + 0.453i)2-s + (−0.707 − 0.707i)3-s + (0.587 − 0.809i)4-s + (0.951 + 0.309i)6-s + (0.996 + 0.0784i)7-s + (−0.156 + 0.987i)8-s + i·9-s + (−0.649 + 0.760i)11-s + (−0.987 + 0.156i)12-s + (0.233 − 0.972i)13-s + (−0.923 + 0.382i)14-s + (−0.309 − 0.951i)16-s + (−0.233 + 0.972i)17-s + (−0.453 − 0.891i)18-s + (0.233 + 0.972i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.981 + 0.191i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (2148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.981 + 0.191i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9861022942 + 0.09512780024i\)
\(L(\frac12)\) \(\approx\) \(0.9861022942 + 0.09512780024i\)
\(L(1)\) \(\approx\) \(0.6654075389 + 0.02718315261i\)
\(L(1)\) \(\approx\) \(0.6654075389 + 0.02718315261i\)

Euler product

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

−17.901879950133518401092409333, −16.955268876439198596649024273470, −16.53553560927537752533352789220, −16.0172330987046394328762730953, −15.246205406851367143066158798384, −14.66227734532288710479134459013, −13.55048718777327361095857318581, −13.094464452264740373552375462, −11.91099847663461424402747997602, −11.43174235871152517472743602767, −11.19994656628217302393108280465, −10.528312975704863007544803698045, −9.73127504329011089945871416148, −9.0195528651029139861198780645, −8.67181278845039166233267417935, −7.65090495254820498935847671459, −7.061786413966407554294795398533, −6.27479800119198103564788976713, −5.371100837644943696453914683417, −4.65666956786925701108825810497, −4.053549829141067564107727869905, −3.02391043472275878807721624055, −2.43112780885022145053458562943, −1.25407013836808289474710336031, −0.6292168653999447326741449240, 0.6539587400328050381328420598, 1.52034497234902311419696944914, 1.99516150020156219837793035347, 2.917380953758303228214547154264, 4.32628380288303400244353431281, 5.16816338525660691620288243292, 5.617451820923256076718025016989, 6.26669836559795027061792509801, 7.20519992220005557261933130486, 7.659820755109132357726686813709, 8.2022117069757432461378373752, 8.77734456223606969368418968555, 10.00344057862302033042823376939, 10.38237394382877081132393875485, 11.07716086314888587412962840385, 11.567808137992304269437395215856, 12.48518999946556778322714628376, 12.94542047225000376923753061759, 13.904679667507642687346923457560, 14.603954083314619552675576625513, 15.38358551611928892173461012011, 15.663814109542209682580105735772, 16.805066140185271972566115938585, 17.252630203685230865164720566585, 17.58328697875761807178218993393

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