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Properties

Label 1-55e2-3025.1446-r0-0-0
Degree $1$
Conductor $3025$
Sign $-0.374 + 0.927i$
Analytic cond. $14.0480$
Root an. cond. $14.0480$
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  + (0.841 − 0.540i)2-s + (−0.809 + 0.587i)3-s + (0.415 − 0.909i)4-s + (−0.362 + 0.931i)6-s + (0.516 + 0.856i)7-s + (−0.142 − 0.989i)8-s + (0.309 − 0.951i)9-s + (0.198 + 0.980i)12-s + (−0.736 − 0.676i)13-s + (0.897 + 0.441i)14-s + (−0.654 − 0.755i)16-s + (0.941 − 0.336i)17-s + (−0.254 − 0.967i)18-s + (−0.959 + 0.281i)19-s + (−0.921 − 0.389i)21-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)2-s + (−0.809 + 0.587i)3-s + (0.415 − 0.909i)4-s + (−0.362 + 0.931i)6-s + (0.516 + 0.856i)7-s + (−0.142 − 0.989i)8-s + (0.309 − 0.951i)9-s + (0.198 + 0.980i)12-s + (−0.736 − 0.676i)13-s + (0.897 + 0.441i)14-s + (−0.654 − 0.755i)16-s + (0.941 − 0.336i)17-s + (−0.254 − 0.967i)18-s + (−0.959 + 0.281i)19-s + (−0.921 − 0.389i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $-0.374 + 0.927i$
Analytic conductor: \(14.0480\)
Root analytic conductor: \(14.0480\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (1446, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3025,\ (0:\ ),\ -0.374 + 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4128945653 + 0.6119890232i\)
\(L(\frac12)\) \(\approx\) \(0.4128945653 + 0.6119890232i\)
\(L(1)\) \(\approx\) \(1.094319728 - 0.09000477292i\)
\(L(1)\) \(\approx\) \(1.094319728 - 0.09000477292i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.841 - 0.540i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (0.516 + 0.856i)T \)
13 \( 1 + (-0.736 - 0.676i)T \)
17 \( 1 + (0.941 - 0.336i)T \)
19 \( 1 + (-0.959 + 0.281i)T \)
23 \( 1 + (0.0855 + 0.996i)T \)
29 \( 1 + (-0.959 + 0.281i)T \)
31 \( 1 + (-0.736 + 0.676i)T \)
37 \( 1 + (-0.736 + 0.676i)T \)
41 \( 1 + (-0.998 - 0.0570i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (-0.998 + 0.0570i)T \)
53 \( 1 + (0.974 - 0.226i)T \)
59 \( 1 + (-0.254 + 0.967i)T \)
61 \( 1 + (-0.254 + 0.967i)T \)
67 \( 1 + (-0.254 - 0.967i)T \)
71 \( 1 + (-0.0285 + 0.999i)T \)
73 \( 1 + (0.974 + 0.226i)T \)
79 \( 1 + (0.696 - 0.717i)T \)
83 \( 1 + (0.0855 - 0.996i)T \)
89 \( 1 + (0.610 + 0.791i)T \)
97 \( 1 + (-0.466 + 0.884i)T \)
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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

−18.74307003561422086820883970458, −17.91124026342677747384205741540, −17.13144490396532713307905027100, −16.74054216978665421970814382599, −16.3777260788180567423585803824, −15.12323441359983636571529248559, −14.58467206618302960776434544069, −13.92333741673716665790176960633, −13.16146613787020052738287881486, −12.581463988466826151863957227111, −11.944055981155674079962680645537, −11.147862379500399509098008642424, −10.68583663060447341027486731218, −9.64217768120323183719970685736, −8.35391916280808057269774350641, −7.80565938392293341573613791599, −7.03877732740806680260346646613, −6.59721791505861302819446986608, −5.70677008036530790598751663104, −4.94857708008750169446501655035, −4.3731777912189362038261932947, −3.562991403169395945313627871784, −2.25887309006933025865384963384, −1.649700858658692221739642534145, −0.173694653478607926734156503058, 1.26517536022197969374410696610, 2.089655395434228293878756338262, 3.18475643234350993574186405652, 3.75330297418636027889302577228, 4.87042267040268884295255170889, 5.30029527173488833735470814941, 5.75607219585219495434836114112, 6.72766700303257651854288906137, 7.57239316687468524101950002594, 8.76725327505737525165942804757, 9.5500052998322563460868104463, 10.28431158122463889939130138528, 10.79929312849365190607866567225, 11.72878988513461317296524545083, 12.059203954719829674564960976976, 12.686322013264939356387968285210, 13.54530901745619358308141747530, 14.66217627936572821157831002611, 14.95155757158851448559379416672, 15.53102482427585262109429546331, 16.41001187916338196022155547664, 17.087578038829729194232632264843, 17.9640622151086464193390789063, 18.58478485985422484836399984714, 19.302162855209622515702814141833

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