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Properties

Label 1-5025-5025.881-r1-0-0
Degree $1$
Conductor $5025$
Sign $-0.829 + 0.558i$
Analytic cond. $540.011$
Root an. cond. $540.011$
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  + (0.625 + 0.780i)2-s + (−0.217 + 0.976i)4-s + (−0.888 − 0.458i)7-s + (−0.897 + 0.441i)8-s + (0.290 + 0.956i)11-s + (0.272 + 0.962i)13-s + (−0.198 − 0.980i)14-s + (−0.905 − 0.424i)16-s + (0.217 + 0.976i)17-s + (−0.710 − 0.703i)19-s + (−0.564 + 0.825i)22-s + (−0.380 − 0.924i)23-s + (−0.580 + 0.814i)26-s + (0.640 − 0.768i)28-s + (0.978 − 0.207i)29-s + ⋯
L(s)  = 1  + (0.625 + 0.780i)2-s + (−0.217 + 0.976i)4-s + (−0.888 − 0.458i)7-s + (−0.897 + 0.441i)8-s + (0.290 + 0.956i)11-s + (0.272 + 0.962i)13-s + (−0.198 − 0.980i)14-s + (−0.905 − 0.424i)16-s + (0.217 + 0.976i)17-s + (−0.710 − 0.703i)19-s + (−0.564 + 0.825i)22-s + (−0.380 − 0.924i)23-s + (−0.580 + 0.814i)26-s + (0.640 − 0.768i)28-s + (0.978 − 0.207i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5025\)    =    \(3 \cdot 5^{2} \cdot 67\)
Sign: $-0.829 + 0.558i$
Analytic conductor: \(540.011\)
Root analytic conductor: \(540.011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5025} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5025,\ (1:\ ),\ -0.829 + 0.558i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7169297235 + 2.346108706i\)
\(L(\frac12)\) \(\approx\) \(0.7169297235 + 2.346108706i\)
\(L(1)\) \(\approx\) \(1.042832024 + 0.7230546714i\)
\(L(1)\) \(\approx\) \(1.042832024 + 0.7230546714i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.625 + 0.780i)T \)
7 \( 1 + (-0.888 - 0.458i)T \)
11 \( 1 + (0.290 + 0.956i)T \)
13 \( 1 + (0.272 + 0.962i)T \)
17 \( 1 + (0.217 + 0.976i)T \)
19 \( 1 + (-0.710 - 0.703i)T \)
23 \( 1 + (-0.380 - 0.924i)T \)
29 \( 1 + (0.978 - 0.207i)T \)
31 \( 1 + (-0.830 - 0.556i)T \)
37 \( 1 + (0.913 - 0.406i)T \)
41 \( 1 + (0.948 - 0.318i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (-0.879 + 0.475i)T \)
53 \( 1 + (0.736 + 0.676i)T \)
59 \( 1 + (0.466 - 0.884i)T \)
61 \( 1 + (0.820 + 0.572i)T \)
71 \( 1 + (0.217 - 0.976i)T \)
73 \( 1 + (0.820 + 0.572i)T \)
79 \( 1 + (-0.0665 - 0.997i)T \)
83 \( 1 + (-0.483 - 0.875i)T \)
89 \( 1 + (-0.941 + 0.336i)T \)
97 \( 1 + (0.669 + 0.743i)T \)
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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

−18.002605574840384825804441145478, −16.79753605716737653514491539437, −16.09789689220109382352181613341, −15.61115458355214834315407503974, −14.82765240281114490491930125415, −14.10163986287648114005837514611, −13.54026859650730488162040155801, −12.811110799846128882018429492487, −12.38866224997304018492406797043, −11.54109770875333155215286233124, −11.0632120177606589858182059650, −10.09736819366452080758103171849, −9.78763572156693715425551404588, −8.84636451501016851416865879932, −8.31464540607262144043714171300, −7.13820001752409901712195395015, −6.32501662027080743204561544301, −5.69603190376130047004481620119, −5.26989674965852357968252015619, −4.113563979853491750301887705049, −3.4351920732553693678429847321, −2.951759344646414844308938444707, −2.14750649846936944375421200309, −1.03259494908742455629632787011, −0.4074704581457919511328593063, 0.67565260511499988226074805592, 2.022824999482488880967770116151, 2.71421859748277118245046464236, 3.82754209970504042069572591807, 4.20270079991150929599718221966, 4.81028533736547394412984036291, 6.120303900121273004876890188049, 6.2985614512455926068495726717, 7.052166651385366883671010022528, 7.69233263010414671287686920910, 8.57452369630089385553198440457, 9.24133108995349238192072432062, 9.89191963022917567872217227424, 10.803176075662748032158015792436, 11.577087855405696353498360354455, 12.54859148846241144463208476158, 12.74755456028945920973937107989, 13.47492156581334970937539536750, 14.34190874458044760250756770824, 14.70189114845223026754803020300, 15.48664397065928069329660607183, 16.23805005942091123645980973843, 16.605689458710588602443074699128, 17.35149319682444895601990849973, 17.87060702502952570588998052798

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