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Properties

Label 1-2736-2736.1165-r0-0-0
Degree $1$
Conductor $2736$
Sign $-0.989 + 0.145i$
Analytic cond. $12.7059$
Root an. cond. $12.7059$
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  + (0.642 − 0.766i)5-s − 7-s + (−0.866 + 0.5i)11-s + (0.342 − 0.939i)13-s + (0.173 − 0.984i)17-s + (−0.173 − 0.984i)23-s + (−0.173 − 0.984i)25-s + (0.342 − 0.939i)29-s + (−0.5 + 0.866i)31-s + (−0.642 + 0.766i)35-s − i·37-s + (−0.173 + 0.984i)41-s + (−0.984 − 0.173i)43-s + (−0.939 − 0.342i)47-s + 49-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)5-s − 7-s + (−0.866 + 0.5i)11-s + (0.342 − 0.939i)13-s + (0.173 − 0.984i)17-s + (−0.173 − 0.984i)23-s + (−0.173 − 0.984i)25-s + (0.342 − 0.939i)29-s + (−0.5 + 0.866i)31-s + (−0.642 + 0.766i)35-s − i·37-s + (−0.173 + 0.984i)41-s + (−0.984 − 0.173i)43-s + (−0.939 − 0.342i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.989 + 0.145i$
Analytic conductor: \(12.7059\)
Root analytic conductor: \(12.7059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2736,\ (0:\ ),\ -0.989 + 0.145i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03769848278 - 0.5153135488i\)
\(L(\frac12)\) \(\approx\) \(-0.03769848278 - 0.5153135488i\)
\(L(1)\) \(\approx\) \(0.8234903149 - 0.2491228171i\)
\(L(1)\) \(\approx\) \(0.8234903149 - 0.2491228171i\)

Euler product

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

−19.438302712649125642835353459351, −18.779149951894890101904468344164, −18.49363028725358719605368701668, −17.47348970825320818173847228488, −16.85543248091722823204666483138, −16.04533988187105971060051177310, −15.49968837784892254465860311588, −14.58774608610107909634062523948, −13.93853659817049257739638866271, −13.18313308560124725851989324489, −12.81513983285383187383385722199, −11.60634897122499275430534835557, −11.02415282885559240704740865346, −10.14770554655135574498985178629, −9.760487479479458528887131625484, −8.8614255963522913287514184327, −8.0296562138752566281078105871, −7.0385969796936226738454975283, −6.44729911939267697118140387733, −5.82941653030580586589470977945, −5.02534147427159871529669952763, −3.64727736820080905776111943524, −3.32299737282247940006667897422, −2.279401047177355463477394792604, −1.50407318680796487343817078141, 0.160417978831033896512252194413, 1.15967557655549835599252571003, 2.41185982344865353849425878262, 2.91923532303834114452276594568, 4.0380275159609760362078944801, 4.9926644128454928903081069344, 5.54497761849891511531977352730, 6.350419987486829909773424705, 7.16348813199597271438870047660, 8.1032748828191848367350957834, 8.758042253414917932426665314432, 9.70543198354618093530474723299, 10.04917509754153676200276259734, 10.81638896853811299729862585274, 12.00772910751562122779624597141, 12.60209560656321797209836609026, 13.18852962131204724837941831515, 13.6613923796572768978215575883, 14.66072208105342381078825040839, 15.55078585002127593136908088065, 16.17465512963732005538265047720, 16.58390277974933662510329564834, 17.615910880318739206358811119839, 18.11410828197980288071349981758, 18.75637532987095735044402976828

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