Properties

Label 1-2736-2736.211-r0-0-0
Degree $1$
Conductor $2736$
Sign $-0.626 - 0.779i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.626 - 0.779i$
Analytic conductor: \(12.7059\)
Root analytic conductor: \(12.7059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2736,\ (0:\ ),\ -0.626 - 0.779i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06388218649 - 0.1334142458i\)
\(L(\frac12)\) \(\approx\) \(0.06388218649 - 0.1334142458i\)
\(L(1)\) \(\approx\) \(0.7606937762 + 0.1080633386i\)
\(L(1)\) \(\approx\) \(0.7606937762 + 0.1080633386i\)

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

−19.599644430805474849035924383136, −18.92403696337160910988674723478, −18.07423867543782950099300145791, −17.42316387673391013725680324614, −16.54231155275820653983348122726, −15.96981663777416559527758997386, −15.42776219229561270290405580327, −14.78012978459056793611107639418, −13.44204962588610551934072871991, −13.212867491632628397921994787659, −12.482871165088720316253996432, −11.611800024208655312787645531179, −11.13990094120607392081476215045, −10.077703996050784777043192738990, −9.24675525345045386110383905006, −8.62154139193004698766595422250, −8.168909127693801322651207508972, −7.1725602021710109996884285749, −6.02627647113105076297115837025, −5.658871793556353656816656731, −4.82255646967821713772962166651, −3.76459206773196688634894180526, −3.160881158314580111561065623121, −2.10523568196314123137051845472, −1.04795828444304055069610551898, 0.05008217764107741226424071084, 1.51531314016559602273617268782, 2.474830131612262779233767158642, 3.25037572259370790253371048222, 4.344714629270453379012801039941, 4.48011988559127090750647445432, 6.06698168959709002951678395598, 6.74375063072075118460468395932, 7.12022259444680870718265222598, 7.94648919422519250084580353977, 8.96982266273597085811071167670, 9.78781464620161757728962621824, 10.36140654366925113246187947143, 11.149788098481303656203180667348, 11.69401872512507512763955463626, 12.66519954347471780529634095003, 13.3928891362910189937746005128, 14.09838179659895273886035488100, 14.71521995148974357691435028122, 15.52190656250461440913953454416, 16.08516036928601375165374652557, 16.976224559220771063033137714203, 17.578679180964067484593333809976, 18.4119168764125624616057660878, 18.995602688229568463621811041693

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