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Properties

Label 1-6025-6025.5827-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.323 - 0.946i$
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 + (−0.891 + 0.453i)3-s + (0.587 − 0.809i)4-s + (0.587 − 0.809i)6-s + (0.522 + 0.852i)7-s + (−0.156 + 0.987i)8-s + (0.587 − 0.809i)9-s + (−0.760 − 0.649i)11-s + (−0.156 + 0.987i)12-s + (0.522 − 0.852i)13-s + (−0.852 − 0.522i)14-s + (−0.309 − 0.951i)16-s + (0.923 − 0.382i)17-s + (−0.156 + 0.987i)18-s + (−0.972 + 0.233i)19-s + ⋯
L(s)  = 1  + (−0.891 + 0.453i)2-s + (−0.891 + 0.453i)3-s + (0.587 − 0.809i)4-s + (0.587 − 0.809i)6-s + (0.522 + 0.852i)7-s + (−0.156 + 0.987i)8-s + (0.587 − 0.809i)9-s + (−0.760 − 0.649i)11-s + (−0.156 + 0.987i)12-s + (0.522 − 0.852i)13-s + (−0.852 − 0.522i)14-s + (−0.309 − 0.951i)16-s + (0.923 − 0.382i)17-s + (−0.156 + 0.987i)18-s + (−0.972 + 0.233i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.323 - 0.946i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (5827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.323 - 0.946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3498895283 - 0.2502546045i\)
\(L(\frac12)\) \(\approx\) \(0.3498895283 - 0.2502546045i\)
\(L(1)\) \(\approx\) \(0.5100375896 + 0.1073770478i\)
\(L(1)\) \(\approx\) \(0.5100375896 + 0.1073770478i\)

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 \( 1 + (-0.891 + 0.453i)T \)
7 \( 1 + (0.522 + 0.852i)T \)
11 \( 1 + (-0.760 - 0.649i)T \)
13 \( 1 + (0.522 - 0.852i)T \)
17 \( 1 + (0.923 - 0.382i)T \)
19 \( 1 + (-0.972 + 0.233i)T \)
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 \( 1 + (-0.453 - 0.891i)T \)
43 \( 1 + (-0.852 - 0.522i)T \)
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 \( 1 + (0.522 + 0.852i)T \)
79 \( 1 + (-0.156 + 0.987i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (-0.996 + 0.0784i)T \)
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

−17.83151040643438853031049709044, −17.22545460817875733798147331859, −16.779782254105633573009865881606, −16.231132184175239510523245778874, −15.48130611497697643046318642732, −14.58451886687443471541609846265, −13.67269802302174391876063860634, −12.95977308016989215073351716117, −12.54344358094553579134557005662, −11.636875797933162614455758510864, −11.21399405024252236174081263080, −10.61382664390427101809839509575, −10.027153245656166631250050734957, −9.40685999700969221840089893513, −8.22527986214481416607066078929, −7.82739807353472664038074494561, −7.2715057196014392615004952171, −6.472305511202659085394379751037, −5.901327703378367924449776069990, −4.71659051390743954222221858766, −4.24167306509232684437457727405, −3.31992176983222117337372474620, −2.05053903976259797433170646061, −1.72844148396214840307109026978, −0.852939964961738612302979461591, 0.22129675200937336083311785801, 1.10658875201673332974168563811, 2.08024570370364173378743320661, 2.93344478266273266183945110184, 3.94713879346409766839187322661, 5.06145195575054170211220002519, 5.51289152268036658828033353110, 5.96716991842301793872839380956, 6.67789224341620604447927183634, 7.69263569571674526161805471409, 8.29345558372968480154713242585, 8.74885079930288498041075047234, 9.75425718457701892364580672710, 10.28144201447777696618221854152, 10.79878178636305429910858843743, 11.49029478147368599951098237654, 12.058808163105014093240750387985, 12.78751908374415694797320851633, 13.79886390355600353559832030153, 14.64271530654336511957200673579, 15.3089299261375992196364784125, 15.664906159500999666261289303697, 16.35575698687611902921769517552, 16.90373353200317651124159652259, 17.57100009484121045895081745863

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