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Properties

Label 1-904-904.459-r1-0-0
Degree $1$
Conductor $904$
Sign $-0.813 + 0.581i$
Analytic cond. $97.1482$
Root an. cond. $97.1482$
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  + (0.900 + 0.433i)3-s + (−0.900 + 0.433i)5-s + (0.900 + 0.433i)7-s + (0.623 + 0.781i)9-s + (−0.222 + 0.974i)11-s + (0.900 + 0.433i)13-s − 15-s + (−0.623 + 0.781i)17-s + (0.900 + 0.433i)19-s + (0.623 + 0.781i)21-s + (−0.900 + 0.433i)23-s + (0.623 − 0.781i)25-s + (0.222 + 0.974i)27-s + (0.623 − 0.781i)29-s + (0.900 + 0.433i)31-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)3-s + (−0.900 + 0.433i)5-s + (0.900 + 0.433i)7-s + (0.623 + 0.781i)9-s + (−0.222 + 0.974i)11-s + (0.900 + 0.433i)13-s − 15-s + (−0.623 + 0.781i)17-s + (0.900 + 0.433i)19-s + (0.623 + 0.781i)21-s + (−0.900 + 0.433i)23-s + (0.623 − 0.781i)25-s + (0.222 + 0.974i)27-s + (0.623 − 0.781i)29-s + (0.900 + 0.433i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 904 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 904 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(904\)    =    \(2^{3} \cdot 113\)
Sign: $-0.813 + 0.581i$
Analytic conductor: \(97.1482\)
Root analytic conductor: \(97.1482\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{904} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 904,\ (1:\ ),\ -0.813 + 0.581i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8495498305 + 2.651640570i\)
\(L(\frac12)\) \(\approx\) \(0.8495498305 + 2.651640570i\)
\(L(1)\) \(\approx\) \(1.245918148 + 0.7262467038i\)
\(L(1)\) \(\approx\) \(1.245918148 + 0.7262467038i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
113 \( 1 \)
good3 \( 1 + (0.900 + 0.433i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + (0.900 + 0.433i)T \)
11 \( 1 + (-0.222 + 0.974i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
17 \( 1 + (-0.623 + 0.781i)T \)
19 \( 1 + (0.900 + 0.433i)T \)
23 \( 1 + (-0.900 + 0.433i)T \)
29 \( 1 + (0.623 - 0.781i)T \)
31 \( 1 + (0.900 + 0.433i)T \)
37 \( 1 + (-0.222 + 0.974i)T \)
41 \( 1 + (-0.222 - 0.974i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (0.222 + 0.974i)T \)
59 \( 1 + (0.900 + 0.433i)T \)
61 \( 1 + (0.222 - 0.974i)T \)
67 \( 1 + (0.222 - 0.974i)T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 + (-0.222 - 0.974i)T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (-0.623 - 0.781i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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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

−21.0386107499895360417405775444, −20.527567293183422041232889282932, −19.93982489579949403077890429657, −19.17921800190281524963694144146, −18.20678873716869269103770513212, −17.81147531875552423841610910162, −16.330395215650557462915862798161, −15.86895632611452197199174679595, −15.0117869490758129473299528374, −13.98425585005087273707789957766, −13.59625815184965741157505034604, −12.64073127384797075424070679441, −11.60480502070584468520741749633, −11.100638042682097735358903193064, −9.8950958643945289417193396431, −8.56126903572513953324044537102, −8.442867916071653242434447202274, −7.55298806242117971209268282942, −6.71779273791547679842844833895, −5.37303771614961417189457839244, −4.34953720832432720254969489373, −3.547849119925283723777756309691, −2.63005278121230084902104019334, −1.23083788960314012977820381140, −0.56460822154053539901781463739, 1.48257444400943018433835625562, 2.34219159455592915220010366305, 3.47853028443936020445916166783, 4.23576850714121213654483349746, 4.97263609162157320133379223073, 6.382077519933475558618843218564, 7.46323490916714189851800254394, 8.16408335557022254785956268177, 8.67928041697790483823758646441, 9.87611416694755973863366912062, 10.582731254295686628865923098266, 11.58328809984912343678353037932, 12.18445742335493412075774220654, 13.46670710649308798178128525903, 14.152062257135643054735347625385, 15.018834055964873721146056681605, 15.509595217264863580468721559761, 16.07845559146485247217245716862, 17.42473081554802817737709460239, 18.291784337689554136196883743482, 18.89398837685224896792612657340, 19.85402496931807606623512664534, 20.38695709367411416944011035165, 21.18184243809255978887395597652, 21.901761331758144271272323104502

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