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Properties

Label 1-1832-1832.1117-r1-0-0
Degree $1$
Conductor $1832$
Sign $-0.605 + 0.795i$
Analytic cond. $196.875$
Root an. cond. $196.875$
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  + (−0.904 + 0.426i)3-s + (−0.962 − 0.272i)5-s + (−0.892 + 0.451i)7-s + (0.635 − 0.771i)9-s + (−0.677 + 0.735i)11-s + (−0.969 + 0.245i)13-s + (0.986 − 0.164i)15-s + (0.245 + 0.969i)17-s + (−0.716 − 0.697i)19-s + (0.614 − 0.789i)21-s + (−0.110 − 0.993i)23-s + (0.851 + 0.523i)25-s + (−0.245 + 0.969i)27-s + (0.0550 − 0.998i)29-s + (−0.954 + 0.298i)31-s + ⋯
L(s)  = 1  + (−0.904 + 0.426i)3-s + (−0.962 − 0.272i)5-s + (−0.892 + 0.451i)7-s + (0.635 − 0.771i)9-s + (−0.677 + 0.735i)11-s + (−0.969 + 0.245i)13-s + (0.986 − 0.164i)15-s + (0.245 + 0.969i)17-s + (−0.716 − 0.697i)19-s + (0.614 − 0.789i)21-s + (−0.110 − 0.993i)23-s + (0.851 + 0.523i)25-s + (−0.245 + 0.969i)27-s + (0.0550 − 0.998i)29-s + (−0.954 + 0.298i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1832\)    =    \(2^{3} \cdot 229\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(196.875\)
Root analytic conductor: \(196.875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1832} (1117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1832,\ (1:\ ),\ -0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09807718645 + 0.1979695682i\)
\(L(\frac12)\) \(\approx\) \(0.09807718645 + 0.1979695682i\)
\(L(1)\) \(\approx\) \(0.4756158786 + 0.06955815916i\)
\(L(1)\) \(\approx\) \(0.4756158786 + 0.06955815916i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (-0.904 + 0.426i)T \)
5 \( 1 + (-0.962 - 0.272i)T \)
7 \( 1 + (-0.892 + 0.451i)T \)
11 \( 1 + (-0.677 + 0.735i)T \)
13 \( 1 + (-0.969 + 0.245i)T \)
17 \( 1 + (0.245 + 0.969i)T \)
19 \( 1 + (-0.716 - 0.697i)T \)
23 \( 1 + (-0.110 - 0.993i)T \)
29 \( 1 + (0.0550 - 0.998i)T \)
31 \( 1 + (-0.954 + 0.298i)T \)
37 \( 1 + (-0.137 + 0.990i)T \)
41 \( 1 + (0.936 + 0.350i)T \)
43 \( 1 + (-0.789 - 0.614i)T \)
47 \( 1 + (0.771 + 0.635i)T \)
53 \( 1 + (0.0825 - 0.996i)T \)
59 \( 1 + (-0.990 + 0.137i)T \)
61 \( 1 + (0.986 + 0.164i)T \)
67 \( 1 + (0.936 - 0.350i)T \)
71 \( 1 + (-0.975 + 0.218i)T \)
73 \( 1 + (0.523 - 0.851i)T \)
79 \( 1 + (-0.0550 - 0.998i)T \)
83 \( 1 + (0.926 + 0.376i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.0275 - 0.999i)T \)
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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

−19.63416429105645719107713944177, −18.82722384982811762470524022776, −18.481822290544551399531852796945, −17.4657770032989797390126515821, −16.614133583068516593829807341683, −16.18107245846929281566297776371, −15.56974190621016144459984840958, −14.51622417679670943002169202806, −13.667801594157376536798109791560, −12.73391638647060109882842176724, −12.3863859165993723865609636168, −11.457827357326243649146710270329, −10.79885472318912535993916144036, −10.19159060072153189426133665270, −9.22591252975675006301263865972, −7.97934220698098788659390996231, −7.38695849486325895764676106567, −6.88526597243574338963118344194, −5.81898239260406093043986742114, −5.18182262480406886138102375088, −4.11217483753549319094630641211, −3.304386764801616904608747809615, −2.360691533752297635457960013821, −0.88761036899966065470599473955, −0.11540712061835305712376671372, 0.47445961970807904899473037801, 2.03867050651188385940364043764, 3.09147675419031219418792229450, 4.14424725976795123745445776128, 4.66889813287435212700227380061, 5.52693759533673935510326536679, 6.496400210263693806693168466025, 7.10635699703155533102917763581, 8.07700975380333204695597033894, 9.01219390374764896048788849209, 9.84411812993229764025366410042, 10.47228402423146041805078909279, 11.2835546430843153205061255624, 12.267424707037176405609951684880, 12.47072064660973454325680693539, 13.18062800580164245743121028871, 14.798167873220765792831281864, 15.16991291999875570543400277299, 15.82645669071075613252558493341, 16.578644130683440265302987842990, 17.09101296040924074680319037413, 17.97628672919605834214530329926, 18.88637933458752118517052847192, 19.37091330323955910438827360715, 20.23207140469291287329386737770

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