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Properties

Label 1-1104-1104.797-r0-0-0
Degree $1$
Conductor $1104$
Sign $0.787 + 0.616i$
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  + (0.989 + 0.142i)5-s + (0.415 + 0.909i)7-s + (0.281 + 0.959i)11-s + (0.909 + 0.415i)13-s + (0.841 − 0.540i)17-s + (0.540 − 0.841i)19-s + (0.959 + 0.281i)25-s + (−0.540 − 0.841i)29-s + (−0.654 − 0.755i)31-s + (0.281 + 0.959i)35-s + (0.989 − 0.142i)37-s + (−0.142 + 0.989i)41-s + (−0.755 − 0.654i)43-s − 47-s + (−0.654 + 0.755i)49-s + ⋯
L(s)  = 1  + (0.989 + 0.142i)5-s + (0.415 + 0.909i)7-s + (0.281 + 0.959i)11-s + (0.909 + 0.415i)13-s + (0.841 − 0.540i)17-s + (0.540 − 0.841i)19-s + (0.959 + 0.281i)25-s + (−0.540 − 0.841i)29-s + (−0.654 − 0.755i)31-s + (0.281 + 0.959i)35-s + (0.989 − 0.142i)37-s + (−0.142 + 0.989i)41-s + (−0.755 − 0.654i)43-s − 47-s + (−0.654 + 0.755i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1104\)    =    \(2^{4} \cdot 3 \cdot 23\)
Sign: $0.787 + 0.616i$
Analytic conductor: \(5.12695\)
Root analytic conductor: \(5.12695\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1104} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1104,\ (0:\ ),\ 0.787 + 0.616i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.053630631 + 0.7086929687i\)
\(L(\frac12)\) \(\approx\) \(2.053630631 + 0.7086929687i\)
\(L(1)\) \(\approx\) \(1.425816002 + 0.2450438489i\)
\(L(1)\) \(\approx\) \(1.425816002 + 0.2450438489i\)

Euler product

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

−21.31081104744615024790322395381, −20.57891319959746699780751944089, −19.98832616526601921730559731278, −18.844376628089213301779058049179, −18.18136647117831100256694581536, −17.4534356104187411915449553901, −16.471273499013454226053009524431, −16.35425325837560419021614158245, −14.75411172176070696888155006529, −14.25556070424331127197150892246, −13.48956566048444603214970972298, −12.90563209920367338875490053314, −11.78144774466544451072266447615, −10.76612256280208356700117097840, −10.36401165386001213129153196924, −9.353214229745915704819268173179, −8.46172691009722496984869977672, −7.718789506552288351284734878099, −6.58867084322412220705288520108, −5.80166559125094416075370769923, −5.12768845302428666446458426983, −3.79650204607077654020146173322, −3.19336201673740700959548253021, −1.61935328208140690696461887789, −1.089075193290405598881609900, 1.31340344374676295717618954854, 2.127958156958154651014643617359, 2.9969819941351741721364970132, 4.30610536376949855259442979251, 5.27454799979518155937613296376, 5.92327989918272177034213648113, 6.83431488529985986049787796913, 7.77537329630455645900158831988, 8.90486875211785699032514402088, 9.45773203668256225425235367676, 10.175458336133512285707346131803, 11.41137029849346732534207505629, 11.81533172611204485820062498239, 13.02929466306102791256499804620, 13.521047324908547726355608965587, 14.641827360167429153417164128430, 14.97791302883131069281525123862, 16.10185021070280398627788230521, 16.85812747064608691628377915924, 17.867368695117092212821522852029, 18.207387008070743502471044627866, 18.940669897458686503724892975650, 20.11694923648961464948786501322, 20.84467282803648522499820062688, 21.411645577755197839946116359454

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