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Properties

Label 1-65e2-4225.621-r0-0-0
Degree $1$
Conductor $4225$
Sign $0.965 + 0.261i$
Analytic cond. $19.6208$
Root an. cond. $19.6208$
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  + (0.414 − 0.910i)2-s + (0.152 + 0.988i)3-s + (−0.657 − 0.753i)4-s + (0.962 + 0.270i)6-s + (−0.428 + 0.903i)7-s + (−0.958 + 0.285i)8-s + (−0.953 + 0.301i)9-s + (0.953 + 0.301i)11-s + (0.644 − 0.764i)12-s + (0.644 + 0.764i)14-s + (−0.136 + 0.990i)16-s + (−0.726 − 0.686i)17-s + (−0.120 + 0.992i)18-s + (0.978 − 0.207i)19-s + (−0.958 − 0.285i)21-s + (0.669 − 0.743i)22-s + ⋯
L(s)  = 1  + (0.414 − 0.910i)2-s + (0.152 + 0.988i)3-s + (−0.657 − 0.753i)4-s + (0.962 + 0.270i)6-s + (−0.428 + 0.903i)7-s + (−0.958 + 0.285i)8-s + (−0.953 + 0.301i)9-s + (0.953 + 0.301i)11-s + (0.644 − 0.764i)12-s + (0.644 + 0.764i)14-s + (−0.136 + 0.990i)16-s + (−0.726 − 0.686i)17-s + (−0.120 + 0.992i)18-s + (0.978 − 0.207i)19-s + (−0.958 − 0.285i)21-s + (0.669 − 0.743i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $0.965 + 0.261i$
Analytic conductor: \(19.6208\)
Root analytic conductor: \(19.6208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4225} (621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4225,\ (0:\ ),\ 0.965 + 0.261i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.902961996 + 0.2535893076i\)
\(L(\frac12)\) \(\approx\) \(1.902961996 + 0.2535893076i\)
\(L(1)\) \(\approx\) \(1.233176160 - 0.08732407821i\)
\(L(1)\) \(\approx\) \(1.233176160 - 0.08732407821i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.414 - 0.910i)T \)
3 \( 1 + (0.152 + 0.988i)T \)
7 \( 1 + (-0.428 + 0.903i)T \)
11 \( 1 + (0.953 + 0.301i)T \)
17 \( 1 + (-0.726 - 0.686i)T \)
19 \( 1 + (0.978 - 0.207i)T \)
23 \( 1 + (0.913 - 0.406i)T \)
29 \( 1 + (0.870 - 0.493i)T \)
31 \( 1 + (-0.715 + 0.698i)T \)
37 \( 1 + (-0.369 + 0.929i)T \)
41 \( 1 + (0.892 - 0.450i)T \)
43 \( 1 + (-0.845 - 0.534i)T \)
47 \( 1 + (0.906 - 0.421i)T \)
53 \( 1 + (-0.607 + 0.794i)T \)
59 \( 1 + (0.984 + 0.176i)T \)
61 \( 1 + (0.759 + 0.650i)T \)
67 \( 1 + (0.657 - 0.753i)T \)
71 \( 1 + (0.932 + 0.362i)T \)
73 \( 1 + (-0.215 - 0.976i)T \)
79 \( 1 + (-0.906 + 0.421i)T \)
83 \( 1 + (-0.779 - 0.626i)T \)
89 \( 1 + (-0.913 + 0.406i)T \)
97 \( 1 + (0.827 - 0.561i)T \)
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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

−18.195300227889293566820653064787, −17.360140867062196196524797308561, −17.20909387281940131326724594813, −16.327579790255543980323083492487, −15.74171191495900997669182410686, −14.65176858728991107943316822525, −14.29278876978829039843059635014, −13.69093890454880770687602278063, −12.895157038406255572029412406968, −12.703322533458783825069438232, −11.584726340392036550539643931508, −11.12282178698950884889115473990, −9.835470346039234518838716453004, −9.13038978809296352348337429371, −8.48201182541517082890795697429, −7.699545349860451180449301689851, −7.04511262423908227621296947324, −6.623909913716237662707763383733, −5.92861783333314690366235846548, −5.130539134977374596718624503165, −4.03388054171007013063312522458, −3.54367663559673212011627943436, −2.7066172018633631668180447265, −1.44351408810329641978360750817, −0.62554321862763176802033220039, 0.80045806563177926777813363309, 2.018911792559271599113769602798, 2.78131472493500190658413480637, 3.31429859961923423577068958238, 4.15564099827182921924655293042, 4.88782480076289598159506745857, 5.40922161318729304769891767066, 6.27468233687094975029075798707, 7.08880569965040409583093736025, 8.64947492393560259120090370630, 8.84663559115925511438228506992, 9.59703618071473692779581090064, 10.05818308866667331619597693485, 10.98429557086317441687611427758, 11.588452121582561904894767392430, 12.07938129493549049849828924427, 12.85507468238751484968230111236, 13.8046126981268720063510742646, 14.2077399518928979114351739283, 15.08324869016698831677207616569, 15.49198219415169020995675059102, 16.17728264048522972169162924907, 17.11749373010146126741446860782, 17.81097813991772798741943477915, 18.5718055040465182702985141810

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