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Properties

Label 1-2736-2736.277-r0-0-0
Degree $1$
Conductor $2736$
Sign $-0.937 + 0.347i$
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  + i·5-s + (0.5 + 0.866i)7-s + (0.866 − 0.5i)11-s + (−0.866 + 0.5i)13-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)23-s − 25-s − i·29-s + (−0.5 + 0.866i)31-s + (−0.866 + 0.5i)35-s + i·37-s − 41-s + (−0.866 − 0.5i)43-s + 47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + i·5-s + (0.5 + 0.866i)7-s + (0.866 − 0.5i)11-s + (−0.866 + 0.5i)13-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)23-s − 25-s − i·29-s + (−0.5 + 0.866i)31-s + (−0.866 + 0.5i)35-s + i·37-s − 41-s + (−0.866 − 0.5i)43-s + 47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.937 + 0.347i$
Analytic conductor: \(12.7059\)
Root analytic conductor: \(12.7059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2736,\ (0:\ ),\ -0.937 + 0.347i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1813964486 + 1.013006968i\)
\(L(\frac12)\) \(\approx\) \(0.1813964486 + 1.013006968i\)
\(L(1)\) \(\approx\) \(0.9131358944 + 0.3654235135i\)
\(L(1)\) \(\approx\) \(0.9131358944 + 0.3654235135i\)

Euler product

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

−19.14266423230978031683973274511, −18.00458357621717340851069900167, −17.39916234436776942322423741881, −16.859692282804633332128116215884, −16.442279019740336770096218139567, −15.24834545641428254092688322409, −14.76797583452773243510401535411, −14.01039369980188325245990285112, −13.12709421902258719411644986538, −12.58223411413793868794196581644, −11.95017548841063278638437801576, −10.99895510444149556371338752482, −10.358846614341700496429673643078, −9.48832540871985110905066615468, −8.83837884807140309300950699217, −8.05170465388088947480106915353, −7.33771083446886433822503911322, −6.576312366456911229758187757580, −5.5603335282170449978856469514, −4.678299289231951567008166089726, −4.30322578395633531543314955144, −3.364768890553515683034346124828, −2.01572684398481897753973611492, −1.40161419009499968447822492451, −0.30898480688321878074763919850, 1.41409329495102622609525652543, 2.30517361609316459131383334385, 2.98468387994403414660711002072, 3.868688427073075852859208576945, 4.88511766450909721143852031799, 5.583241008057707181203423077787, 6.57316022409149774963263540244, 7.01482438345452119761555439830, 7.92828989118077549977699539135, 8.82656524418142477407276904520, 9.45205850058408631947444595593, 10.190265923358635554990648693984, 11.28160076345619068543850096804, 11.586085999878642416960960952972, 12.1677488079408396350702096759, 13.37244558750088635457895622580, 14.05958840231504549012280919793, 14.58455056607314177108432937515, 15.31693097907358542260119797662, 15.81767793469052139483017651581, 17.01013805771731222929714146205, 17.385645660321903868940604968807, 18.36499217398060750658262972495, 18.7616765541798821634866465252, 19.4463756717558589569548713265

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