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Properties

Label 1-2695-2695.829-r1-0-0
Degree $1$
Conductor $2695$
Sign $0.416 + 0.909i$
Analytic cond. $289.617$
Root an. cond. $289.617$
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  + (0.772 − 0.635i)2-s + (0.525 − 0.850i)3-s + (0.193 − 0.981i)4-s + (−0.134 − 0.990i)6-s + (−0.473 − 0.880i)8-s + (−0.447 − 0.894i)9-s + (−0.733 − 0.680i)12-s + (0.936 + 0.351i)13-s + (−0.925 − 0.379i)16-s + (0.0149 − 0.999i)17-s + (−0.913 − 0.406i)18-s + (−0.913 + 0.406i)19-s + (−0.955 − 0.294i)23-s + (−0.998 − 0.0598i)24-s + (0.946 − 0.323i)26-s + (−0.995 − 0.0896i)27-s + ⋯
L(s)  = 1  + (0.772 − 0.635i)2-s + (0.525 − 0.850i)3-s + (0.193 − 0.981i)4-s + (−0.134 − 0.990i)6-s + (−0.473 − 0.880i)8-s + (−0.447 − 0.894i)9-s + (−0.733 − 0.680i)12-s + (0.936 + 0.351i)13-s + (−0.925 − 0.379i)16-s + (0.0149 − 0.999i)17-s + (−0.913 − 0.406i)18-s + (−0.913 + 0.406i)19-s + (−0.955 − 0.294i)23-s + (−0.998 − 0.0598i)24-s + (0.946 − 0.323i)26-s + (−0.995 − 0.0896i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.416 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.416 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2695\)    =    \(5 \cdot 7^{2} \cdot 11\)
Sign: $0.416 + 0.909i$
Analytic conductor: \(289.617\)
Root analytic conductor: \(289.617\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2695} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2695,\ (1:\ ),\ 0.416 + 0.909i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.033857904 - 0.6637627129i\)
\(L(\frac12)\) \(\approx\) \(-1.033857904 - 0.6637627129i\)
\(L(1)\) \(\approx\) \(1.002582423 - 1.190536603i\)
\(L(1)\) \(\approx\) \(1.002582423 - 1.190536603i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.772 - 0.635i)T \)
3 \( 1 + (0.525 - 0.850i)T \)
13 \( 1 + (0.936 + 0.351i)T \)
17 \( 1 + (0.0149 - 0.999i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (-0.955 - 0.294i)T \)
29 \( 1 + (-0.393 - 0.919i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (-0.992 - 0.119i)T \)
41 \( 1 + (-0.473 - 0.880i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (0.251 - 0.967i)T \)
53 \( 1 + (0.873 + 0.486i)T \)
59 \( 1 + (0.999 - 0.0299i)T \)
61 \( 1 + (0.873 - 0.486i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.995 - 0.0896i)T \)
73 \( 1 + (0.251 + 0.967i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (0.936 - 0.351i)T \)
89 \( 1 + (0.988 - 0.149i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−19.90485500926312017650740491118, −19.15700324677256633626366974037, −18.07289259928121974871655225214, −17.35742841241389301542133535773, −16.59572663441578368874074576085, −16.01647912117484648920248305412, −15.38577211747160371341896244463, −14.76531573226788990470700112327, −14.24972610979669801133956073032, −13.27525184286656556199385612204, −12.98081936655008692357324169180, −11.90130680683688020433785227656, −10.996486981029518766420112998361, −10.52427697152778407240908860396, −9.42345155104576328960161614980, −8.594703250549281504583610718300, −8.19384274030480974607730893985, −7.31770336100430717940679460771, −6.26304896153029013982638826721, −5.691581746302714780022094766466, −4.85009354404187268978928164109, −3.92686730430007740295753054131, −3.64262745204579631880496517023, −2.6135967249302690473235501718, −1.72542142136180590164079457540, 0.1272356017073932005473806868, 1.0343109276681747899967654093, 2.01838546164072903820991279939, 2.4444057245502315966565962724, 3.67893494560900610352968270717, 3.94815699188264426254307188260, 5.25920541643789041632545372107, 5.971563371751322799624177487819, 6.71827417123561369400151479073, 7.37409477630648199285502574862, 8.52750744665238241778618943405, 9.01108852937130935394809189258, 10.01858881172388653102404587207, 10.748771022824356634246366940119, 11.715083782182881238733045291311, 12.06640097411646086736866082829, 12.95050108170901614985825687337, 13.50411923449135129660113707146, 14.13631144811140365159192751829, 14.627804043942390330284569524902, 15.586653629310496868627627444992, 16.16146079596490533392018075116, 17.33303078030713284078290327280, 18.147379851849774215555333482773, 18.83873491582288905767584593695

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