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Properties

Label 1-2808-2808.1195-r1-0-0
Degree $1$
Conductor $2808$
Sign $-0.549 - 0.835i$
Analytic cond. $301.761$
Root an. cond. $301.761$
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 + (0.173 + 0.984i)7-s + (0.939 + 0.342i)11-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.173 + 0.984i)23-s + (0.766 − 0.642i)25-s + (−0.766 + 0.642i)29-s + (0.173 − 0.984i)31-s + (−0.5 − 0.866i)35-s + (−0.5 + 0.866i)37-s + (−0.766 − 0.642i)41-s + (−0.939 − 0.342i)43-s + (0.173 + 0.984i)47-s + (−0.939 + 0.342i)49-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)5-s + (0.173 + 0.984i)7-s + (0.939 + 0.342i)11-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.173 + 0.984i)23-s + (0.766 − 0.642i)25-s + (−0.766 + 0.642i)29-s + (0.173 − 0.984i)31-s + (−0.5 − 0.866i)35-s + (−0.5 + 0.866i)37-s + (−0.766 − 0.642i)41-s + (−0.939 − 0.342i)43-s + (0.173 + 0.984i)47-s + (−0.939 + 0.342i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2808\)    =    \(2^{3} \cdot 3^{3} \cdot 13\)
Sign: $-0.549 - 0.835i$
Analytic conductor: \(301.761\)
Root analytic conductor: \(301.761\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2808} (1195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2808,\ (1:\ ),\ -0.549 - 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2973463917 + 0.5514633454i\)
\(L(\frac12)\) \(\approx\) \(-0.2973463917 + 0.5514633454i\)
\(L(1)\) \(\approx\) \(0.7795301593 + 0.3294099078i\)
\(L(1)\) \(\approx\) \(0.7795301593 + 0.3294099078i\)

Euler product

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

−18.61299999109784249229961016605, −17.81829968647391941073866033963, −17.06362878681238885839235726906, −16.39990452359628121917688500751, −15.9453924421816927416077114340, −15.00051358559763271000327105928, −14.35816115494341869748076925444, −13.57152859595409368850999897985, −13.00276672891383643472358324324, −11.87093283155938832468895864760, −11.58657239304217026080518248392, −10.80963853206755825535934971641, −9.992245975158663516014588622518, −8.98101634573892801573546117263, −8.55946183721524575214570150841, −7.489627044588111163370485780505, −7.07621875661107870674742680704, −6.27967676731097583546547025607, −4.980184026150069531186644604429, −4.512488241309796623786430045513, −3.70034567107083636832810755758, −3.01521323322363706270035759351, −1.685412630039655622169517638759, −0.72717118062560220521535722574, −0.1348091862223839618862154114, 1.37224754036341402189558609410, 2.05926283714417016573108830287, 3.2773403491672611603923122545, 3.77072393836101803040031623114, 4.67909495494007761799617386387, 5.60301341480080040549322009701, 6.37563951013797684471123938355, 7.146662971014204212645447246527, 7.99637245078526214646134525619, 8.56889826338097848254378077946, 9.39102495582877931342632849219, 10.133624662605103736072178831452, 11.21746598235940511427147241472, 11.6107698644906000150377698050, 12.28023800048750733024158781759, 12.91288077264509796685410191612, 14.05167771729970124792325242247, 14.66522979588942426526624971860, 15.36297940989450708322018578080, 15.68864768511303373131280406629, 16.74383695514288597457334282770, 17.34355507444260376277731916702, 18.25156718894956613719965147554, 18.85510093093096427623021419398, 19.360560034550711008366869131122

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