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Properties

Label 1-1368-1368.803-r0-0-0
Degree $1$
Conductor $1368$
Sign $0.211 - 0.977i$
Analytic cond. $6.35296$
Root an. cond. $6.35296$
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  + (−0.939 − 0.342i)5-s − 7-s + (0.5 − 0.866i)11-s + (−0.173 + 0.984i)13-s + (−0.766 + 0.642i)17-s + (0.766 + 0.642i)23-s + (0.766 + 0.642i)25-s + (0.173 − 0.984i)29-s + (0.5 + 0.866i)31-s + (0.939 + 0.342i)35-s − 37-s + (−0.766 + 0.642i)41-s + (0.766 − 0.642i)43-s + (0.173 − 0.984i)47-s + 49-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)5-s − 7-s + (0.5 − 0.866i)11-s + (−0.173 + 0.984i)13-s + (−0.766 + 0.642i)17-s + (0.766 + 0.642i)23-s + (0.766 + 0.642i)25-s + (0.173 − 0.984i)29-s + (0.5 + 0.866i)31-s + (0.939 + 0.342i)35-s − 37-s + (−0.766 + 0.642i)41-s + (0.766 − 0.642i)43-s + (0.173 − 0.984i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.211 - 0.977i$
Analytic conductor: \(6.35296\)
Root analytic conductor: \(6.35296\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (803, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1368,\ (0:\ ),\ 0.211 - 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5885467655 - 0.4746176896i\)
\(L(\frac12)\) \(\approx\) \(0.5885467655 - 0.4746176896i\)
\(L(1)\) \(\approx\) \(0.7534957371 - 0.08950212464i\)
\(L(1)\) \(\approx\) \(0.7534957371 - 0.08950212464i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 - T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (0.173 - 0.984i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (-0.173 - 0.984i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 - T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (-0.939 - 0.342i)T \)
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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

−20.752259928127068870251244667522, −20.14854649824890912765677770969, −19.558647159235574847352727755809, −18.87513976791394890501685813271, −18.05309469760311065641970120740, −17.23408109088431397156464004849, −16.35597350222525250241645322215, −15.55721811039648196803960973435, −15.16040078457614062284331326462, −14.25273484860708781408209803146, −13.20495306208309794541611987540, −12.49100246770385846693953097632, −11.93409834229537567572776504647, −10.88009349468074333478853691568, −10.26266567963319709872469708296, −9.29757932698073556860344000219, −8.57741185047933309475944020153, −7.39764714633480076346017283572, −7.00717523358651759294834240650, −6.10903321144869610068709824689, −4.89353572311216822891688263906, −4.12435775951456601083980566433, −3.15392109098341076717078754254, −2.51362799196881389694341573839, −0.88286160982774527202882991942, 0.3858255541244019375360621648, 1.67672874781976562692652878770, 3.03667213343647979007750523446, 3.743892220244566929703023756011, 4.478342120969883479943783999214, 5.60063713104133019258447972105, 6.65089078056092477976903784424, 7.0739639067158783442957962221, 8.39332002725553871803245254513, 8.825660312015900445343691267187, 9.70181068387095923678488008556, 10.72260761408567366513297417028, 11.58418705350410449855738908002, 12.08224834662026676069609842931, 13.075050577517661382598933672946, 13.65699409382386613696093096655, 14.66687360511359110838596533047, 15.58615849894228193835671670932, 16.04770253734994578885747449055, 16.83937653066406833734633105727, 17.441029694993242194205107376267, 18.88643084125936876278957441353, 19.19158809299772617046437071587, 19.67781760144929348420939552940, 20.61121696736037522247434024316

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