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Properties

Label 1-2736-2736.131-r0-0-0
Degree $1$
Conductor $2736$
Sign $-0.989 + 0.146i$
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  + (0.342 − 0.939i)5-s + (−0.5 − 0.866i)7-s − i·11-s + (−0.342 − 0.939i)13-s + (0.939 + 0.342i)17-s + (−0.173 + 0.984i)23-s + (−0.766 − 0.642i)25-s + (−0.984 − 0.173i)29-s − 31-s + (−0.984 + 0.173i)35-s − i·37-s + (0.766 − 0.642i)41-s + (−0.984 + 0.173i)43-s + (0.173 − 0.984i)47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)5-s + (−0.5 − 0.866i)7-s − i·11-s + (−0.342 − 0.939i)13-s + (0.939 + 0.342i)17-s + (−0.173 + 0.984i)23-s + (−0.766 − 0.642i)25-s + (−0.984 − 0.173i)29-s − 31-s + (−0.984 + 0.173i)35-s − i·37-s + (0.766 − 0.642i)41-s + (−0.984 + 0.173i)43-s + (0.173 − 0.984i)47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.989 + 0.146i$
Analytic conductor: \(12.7059\)
Root analytic conductor: \(12.7059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2736,\ (0:\ ),\ -0.989 + 0.146i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07110667928 - 0.9676418273i\)
\(L(\frac12)\) \(\approx\) \(-0.07110667928 - 0.9676418273i\)
\(L(1)\) \(\approx\) \(0.8272125801 - 0.4469355586i\)
\(L(1)\) \(\approx\) \(0.8272125801 - 0.4469355586i\)

Euler product

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

−19.35283670945163544865356305062, −18.89757979506091885488031403404, −18.27823657595182858342975567623, −17.73462649886780559694428756736, −16.67128815932967528968655082123, −16.23908263131712179948304714797, −15.11118716398568192600427668257, −14.77652563515234561892375918246, −14.167675285301261528774428728500, −13.17429897771165796846784518302, −12.471962038755936146388636203111, −11.80510163646172044023037973317, −11.105341135313838620607429829674, −10.068242462385773414504979082129, −9.67486968353736686349439694500, −8.99869279280729088331336397984, −7.90334901184802282141940372111, −7.08451070594683468345869205029, −6.55682305804504294594317449539, −5.74188104975144899837503957415, −4.94650245593725067665547015482, −3.93960895530803146568930311407, −3.01394150809982308612128621935, −2.32801306093127567713941709509, −1.60666804539492888838504830448, 0.31006475525073663125497493607, 1.11402996434325602335923275151, 2.11276464682776126422662639787, 3.49106425319289055199506898872, 3.687639051687651740991823454549, 4.99108992580015485185153294769, 5.61643577634456311295209482329, 6.20305651280917358450993371115, 7.49452990084145545613442896338, 7.8234117307003541799905312178, 8.85176377497445271239415721401, 9.50489999501971888427154883625, 10.22477196784710604156355549307, 10.898472961199559188065341895812, 11.80952592915038370088030917360, 12.73858504568123286294692669780, 13.086484406175605408117170529040, 13.83840361666050330602076982940, 14.50228104655472470686826790075, 15.52858223962600734852297982415, 16.23751521841847512494625964224, 16.80660823114263434409452319553, 17.27037200183296592846223348244, 18.10562970075559252401155087738, 19.039689724978187790218953672037

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