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Properties

Label 1-1625-1625.67-r0-0-0
Degree $1$
Conductor $1625$
Sign $0.709 + 0.705i$
Analytic cond. $7.54646$
Root an. cond. $7.54646$
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  + (0.228 + 0.973i)2-s + (0.0418 + 0.999i)3-s + (−0.895 + 0.444i)4-s + (−0.963 + 0.268i)6-s + (0.978 − 0.207i)7-s + (−0.637 − 0.770i)8-s + (−0.996 + 0.0836i)9-s + (−0.973 + 0.228i)11-s + (−0.481 − 0.876i)12-s + (0.425 + 0.904i)14-s + (0.604 − 0.796i)16-s + (−0.553 − 0.832i)17-s + (−0.309 − 0.951i)18-s + (0.0418 − 0.999i)19-s + (0.248 + 0.968i)21-s + (−0.444 − 0.895i)22-s + ⋯
L(s)  = 1  + (0.228 + 0.973i)2-s + (0.0418 + 0.999i)3-s + (−0.895 + 0.444i)4-s + (−0.963 + 0.268i)6-s + (0.978 − 0.207i)7-s + (−0.637 − 0.770i)8-s + (−0.996 + 0.0836i)9-s + (−0.973 + 0.228i)11-s + (−0.481 − 0.876i)12-s + (0.425 + 0.904i)14-s + (0.604 − 0.796i)16-s + (−0.553 − 0.832i)17-s + (−0.309 − 0.951i)18-s + (0.0418 − 0.999i)19-s + (0.248 + 0.968i)21-s + (−0.444 − 0.895i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.709 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.709 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1625\)    =    \(5^{3} \cdot 13\)
Sign: $0.709 + 0.705i$
Analytic conductor: \(7.54646\)
Root analytic conductor: \(7.54646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1625} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1625,\ (0:\ ),\ 0.709 + 0.705i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.125358929 + 0.4643259679i\)
\(L(\frac12)\) \(\approx\) \(1.125358929 + 0.4643259679i\)
\(L(1)\) \(\approx\) \(0.8216160214 + 0.6051265918i\)
\(L(1)\) \(\approx\) \(0.8216160214 + 0.6051265918i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.228 + 0.973i)T \)
3 \( 1 + (0.0418 + 0.999i)T \)
7 \( 1 + (0.978 - 0.207i)T \)
11 \( 1 + (-0.973 + 0.228i)T \)
17 \( 1 + (-0.553 - 0.832i)T \)
19 \( 1 + (0.0418 - 0.999i)T \)
23 \( 1 + (0.653 - 0.756i)T \)
29 \( 1 + (-0.783 + 0.621i)T \)
31 \( 1 + (0.998 - 0.0627i)T \)
37 \( 1 + (-0.604 + 0.796i)T \)
41 \( 1 + (0.328 - 0.944i)T \)
43 \( 1 + (-0.994 - 0.104i)T \)
47 \( 1 + (0.637 - 0.770i)T \)
53 \( 1 + (-0.248 - 0.968i)T \)
59 \( 1 + (-0.518 + 0.855i)T \)
61 \( 1 + (0.944 - 0.328i)T \)
67 \( 1 + (0.146 - 0.989i)T \)
71 \( 1 + (0.166 - 0.985i)T \)
73 \( 1 + (0.876 + 0.481i)T \)
79 \( 1 + (-0.535 - 0.844i)T \)
83 \( 1 + (-0.535 + 0.844i)T \)
89 \( 1 + (0.518 + 0.855i)T \)
97 \( 1 + (0.146 + 0.989i)T \)
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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

−20.36277185282187054259811675274, −19.57309027842989284601747865086, −18.743777684105711115310233228833, −18.47840794669166105517196220609, −17.472775171013376619140639427115, −17.19561711160560073622162102962, −15.64488855550022076936388695286, −14.79640075195249660647389342073, −14.15487886441501457870412809299, −13.36016331920458541619923203943, −12.83209937498364906316432744676, −12.04993136409653936388091184580, −11.29597402203981026902082420459, −10.80499252825126839761510842694, −9.797699623122999237747822621837, −8.68590252831243312313357903304, −8.17679473069955545651608613267, −7.445401776245364956589206581724, −6.06953761052044322679911443455, −5.51844501203484346613853048155, −4.61058222239196160861143817545, −3.52590509061739265957805983813, −2.55036195661181973156819068595, −1.85295813419445523412168707330, −1.0693691423041302767868169151, 0.457593746353974662411451145716, 2.35362934786647840108141934148, 3.28149563639606086992011150348, 4.36561419568998853644126004759, 5.0001109610377053687533718637, 5.3115269535015639640391031296, 6.636772329810454385922599124638, 7.39052420880731702041820486172, 8.340100790196262467385281945876, 8.82834978847152150261484089603, 9.73413319695939689759859652123, 10.61112157835387008739329262367, 11.31547492182329521608776327176, 12.278244185980090574366808707023, 13.45257844698528107413060990721, 13.84511821565747807729511951610, 14.86114098750552085408223929489, 15.26790646567772384074574955696, 15.90858220073842661761088764623, 16.73101385213694926691987861215, 17.387152824660873979524364182778, 18.02559769129579580738472394133, 18.7814102301246388320810426440, 20.06733267526637688500931001111, 20.770888302657405369458109413530

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