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Properties

Label 1-2496-2496.1811-r0-0-0
Degree $1$
Conductor $2496$
Sign $0.0852 + 0.996i$
Analytic cond. $11.5913$
Root an. cond. $11.5913$
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  + (−0.923 + 0.382i)5-s + (−0.258 − 0.965i)7-s + (−0.608 + 0.793i)11-s + (0.866 + 0.5i)17-s + (−0.793 + 0.608i)19-s + (0.258 − 0.965i)23-s + (0.707 − 0.707i)25-s + (0.991 − 0.130i)29-s − 31-s + (0.608 + 0.793i)35-s + (0.793 + 0.608i)37-s + (0.258 − 0.965i)41-s + (−0.991 − 0.130i)43-s − i·47-s + (−0.866 + 0.5i)49-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)5-s + (−0.258 − 0.965i)7-s + (−0.608 + 0.793i)11-s + (0.866 + 0.5i)17-s + (−0.793 + 0.608i)19-s + (0.258 − 0.965i)23-s + (0.707 − 0.707i)25-s + (0.991 − 0.130i)29-s − 31-s + (0.608 + 0.793i)35-s + (0.793 + 0.608i)37-s + (0.258 − 0.965i)41-s + (−0.991 − 0.130i)43-s − i·47-s + (−0.866 + 0.5i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0852 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0852 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $0.0852 + 0.996i$
Analytic conductor: \(11.5913\)
Root analytic conductor: \(11.5913\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (1811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2496,\ (0:\ ),\ 0.0852 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5625854641 + 0.5165066416i\)
\(L(\frac12)\) \(\approx\) \(0.5625854641 + 0.5165066416i\)
\(L(1)\) \(\approx\) \(0.7798981399 + 0.06069484365i\)
\(L(1)\) \(\approx\) \(0.7798981399 + 0.06069484365i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.923 + 0.382i)T \)
7 \( 1 + (-0.258 - 0.965i)T \)
11 \( 1 + (-0.608 + 0.793i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.793 + 0.608i)T \)
23 \( 1 + (0.258 - 0.965i)T \)
29 \( 1 + (0.991 - 0.130i)T \)
31 \( 1 - T \)
37 \( 1 + (0.793 + 0.608i)T \)
41 \( 1 + (0.258 - 0.965i)T \)
43 \( 1 + (-0.991 - 0.130i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.382 - 0.923i)T \)
59 \( 1 + (-0.130 + 0.991i)T \)
61 \( 1 + (-0.608 - 0.793i)T \)
67 \( 1 + (0.991 - 0.130i)T \)
71 \( 1 + (0.258 + 0.965i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + iT \)
83 \( 1 + (-0.923 - 0.382i)T \)
89 \( 1 + (-0.965 - 0.258i)T \)
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

−19.40625157212194754479264440506, −18.58177412968366338673625561938, −18.120961768665051014587373828810, −16.99106281784520877487534495739, −16.29308202489198161587502262990, −15.77750956656676864354828528920, −15.16311203887966501642794553014, −14.41682538974469477093227910911, −13.40269195980159802313512245970, −12.72626713900097141621600411048, −12.14056697275280920954035510749, −11.32107417221122852607407633254, −10.85807685040180936524950636160, −9.62697182554464179032889496549, −9.05466146632031538882590124259, −8.22259088166856791450308146273, −7.74242409194204067057849379612, −6.73590708696306367339392049996, −5.79606805146777973143623039740, −5.1573832736689164882846239683, −4.33633883525016215361458341526, −3.20249637929171175279993496327, −2.83454136603817330285437303561, −1.4971801191846432878575796439, −0.317487778119305351477409536743, 0.86942545342604169874463216667, 2.09286717405906945154587051321, 3.09693371397459800482098915347, 3.8972844015600800453285683538, 4.45356122260351117694166372548, 5.42311222108433675839741535712, 6.641255392172822597299328042691, 7.00557721350430487620220836552, 8.01450244352140488123549129426, 8.28713577775012484105765178737, 9.64378780194271281933373722934, 10.406916954024469559559754995778, 10.70262151844371509192785195785, 11.73018282279133675202003793442, 12.54713560283873923182332312495, 12.94647201819197516461251970241, 14.06903780025061071930387523363, 14.67890177411580959326428276986, 15.275779056186315768867180029918, 16.13033602523333130568165781442, 16.740915922010287166864241414988, 17.374462521895786460480656754669, 18.50176234241499532864114005431, 18.75788929824575726816994959572, 19.81614228844524187075511062734

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