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Properties

Label 1-1104-1104.131-r0-0-0
Degree $1$
Conductor $1104$
Sign $0.971 + 0.235i$
Analytic cond. $5.12695$
Root an. cond. $5.12695$
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  + (−0.755 − 0.654i)5-s + (0.841 − 0.540i)7-s + (0.989 + 0.142i)11-s + (−0.540 + 0.841i)13-s + (0.959 + 0.281i)17-s + (0.281 + 0.959i)19-s + (0.142 + 0.989i)25-s + (−0.281 + 0.959i)29-s + (−0.415 − 0.909i)31-s + (−0.989 − 0.142i)35-s + (−0.755 + 0.654i)37-s + (−0.654 + 0.755i)41-s + (0.909 + 0.415i)43-s + 47-s + (0.415 − 0.909i)49-s + ⋯
L(s)  = 1  + (−0.755 − 0.654i)5-s + (0.841 − 0.540i)7-s + (0.989 + 0.142i)11-s + (−0.540 + 0.841i)13-s + (0.959 + 0.281i)17-s + (0.281 + 0.959i)19-s + (0.142 + 0.989i)25-s + (−0.281 + 0.959i)29-s + (−0.415 − 0.909i)31-s + (−0.989 − 0.142i)35-s + (−0.755 + 0.654i)37-s + (−0.654 + 0.755i)41-s + (0.909 + 0.415i)43-s + 47-s + (0.415 − 0.909i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1104\)    =    \(2^{4} \cdot 3 \cdot 23\)
Sign: $0.971 + 0.235i$
Analytic conductor: \(5.12695\)
Root analytic conductor: \(5.12695\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1104} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1104,\ (0:\ ),\ 0.971 + 0.235i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.485797014 + 0.1777418683i\)
\(L(\frac12)\) \(\approx\) \(1.485797014 + 0.1777418683i\)
\(L(1)\) \(\approx\) \(1.100777829 + 0.02077761537i\)
\(L(1)\) \(\approx\) \(1.100777829 + 0.02077761537i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 \)
good5 \( 1 + (-0.755 - 0.654i)T \)
7 \( 1 + (0.841 - 0.540i)T \)
11 \( 1 + (0.989 + 0.142i)T \)
13 \( 1 + (-0.540 + 0.841i)T \)
17 \( 1 + (0.959 + 0.281i)T \)
19 \( 1 + (0.281 + 0.959i)T \)
29 \( 1 + (-0.281 + 0.959i)T \)
31 \( 1 + (-0.415 - 0.909i)T \)
37 \( 1 + (-0.755 + 0.654i)T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (0.909 + 0.415i)T \)
47 \( 1 + T \)
53 \( 1 + (0.540 + 0.841i)T \)
59 \( 1 + (0.540 - 0.841i)T \)
61 \( 1 + (-0.909 + 0.415i)T \)
67 \( 1 + (-0.989 + 0.142i)T \)
71 \( 1 + (0.142 + 0.989i)T \)
73 \( 1 + (0.959 - 0.281i)T \)
79 \( 1 + (-0.841 - 0.540i)T \)
83 \( 1 + (0.755 - 0.654i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
97 \( 1 + (-0.654 + 0.755i)T \)
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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

−21.426868828425315718560891678537, −20.57575230178556633234007774602, −19.66093401161759577621272602662, −19.16870885401208142593551279741, −18.23907400274260176424650875703, −17.61255171660206043315820118436, −16.786089864483352226304540502974, −15.67034730099168000114859130122, −15.14931997010192637583031823893, −14.41866567517001110411012286965, −13.7853045255655261841184892242, −12.32019586544201930758680788703, −11.982955500520237978066235671691, −11.12375823915363594988123111758, −10.41161780269129054883967960494, −9.295835498800702179599281587563, −8.48915235982685736683731019144, −7.575052075264035645048934860000, −7.02986998084642840400301901268, −5.80822159432223769919970929093, −5.02249177294620422664470977319, −3.95990782570381031225761213394, −3.099930435428243452916014375381, −2.13862940714731580565684428376, −0.76030979054320357468615807033, 1.106507381549805530165094620437, 1.7721776345486093825707119269, 3.44774347448672037919504650338, 4.154216502263057200132843742918, 4.8617050316465626542894656629, 5.87845241633014795272801606296, 7.14474283519686099568267839394, 7.67113116637098023489756751536, 8.57055888113436652673681386594, 9.35745893375620010543612135314, 10.30865090181076043831930538989, 11.35595734783797729419420356138, 11.94463095770590045002448683382, 12.52733815110110218419246972047, 13.7313433596979369256094282839, 14.497758378243174602142029249962, 14.97535278531296098573681276111, 16.215440228629766587275472694068, 16.83482960858296455855691531650, 17.2313096023857885523826518256, 18.49623321487827065611283491996, 19.13752272812719282997695017765, 20.01898043320393497333119323549, 20.53183113709486219313701361085, 21.30214438742548018801770126416

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