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Properties

Label 1-517-517.112-r1-0-0
Degree $1$
Conductor $517$
Sign $0.217 - 0.976i$
Analytic cond. $55.5593$
Root an. cond. $55.5593$
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  + (0.282 − 0.959i)2-s + (0.994 + 0.109i)3-s + (−0.839 − 0.542i)4-s + (−0.531 − 0.847i)5-s + (0.385 − 0.922i)6-s + (0.955 + 0.295i)7-s + (−0.758 + 0.652i)8-s + (0.976 + 0.216i)9-s + (−0.962 + 0.269i)10-s + (−0.775 − 0.631i)12-s + (0.981 − 0.190i)13-s + (0.554 − 0.832i)14-s + (−0.435 − 0.900i)15-s + (0.410 + 0.911i)16-s + (0.894 + 0.447i)17-s + (0.484 − 0.874i)18-s + ⋯
L(s)  = 1  + (0.282 − 0.959i)2-s + (0.994 + 0.109i)3-s + (−0.839 − 0.542i)4-s + (−0.531 − 0.847i)5-s + (0.385 − 0.922i)6-s + (0.955 + 0.295i)7-s + (−0.758 + 0.652i)8-s + (0.976 + 0.216i)9-s + (−0.962 + 0.269i)10-s + (−0.775 − 0.631i)12-s + (0.981 − 0.190i)13-s + (0.554 − 0.832i)14-s + (−0.435 − 0.900i)15-s + (0.410 + 0.911i)16-s + (0.894 + 0.447i)17-s + (0.484 − 0.874i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.217 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.217 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $0.217 - 0.976i$
Analytic conductor: \(55.5593\)
Root analytic conductor: \(55.5593\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (1:\ ),\ 0.217 - 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.903220656 - 2.328696969i\)
\(L(\frac12)\) \(\approx\) \(2.903220656 - 2.328696969i\)
\(L(1)\) \(\approx\) \(1.577484053 - 0.9129233636i\)
\(L(1)\) \(\approx\) \(1.577484053 - 0.9129233636i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.282 - 0.959i)T \)
3 \( 1 + (0.994 + 0.109i)T \)
5 \( 1 + (-0.531 - 0.847i)T \)
7 \( 1 + (0.955 + 0.295i)T \)
13 \( 1 + (0.981 - 0.190i)T \)
17 \( 1 + (0.894 + 0.447i)T \)
19 \( 1 + (0.969 + 0.243i)T \)
23 \( 1 + (-0.334 + 0.942i)T \)
29 \( 1 + (0.484 - 0.874i)T \)
31 \( 1 + (-0.740 + 0.672i)T \)
37 \( 1 + (0.554 + 0.832i)T \)
41 \( 1 + (0.230 + 0.973i)T \)
43 \( 1 + (0.775 - 0.631i)T \)
53 \( 1 + (-0.996 - 0.0818i)T \)
59 \( 1 + (0.360 - 0.932i)T \)
61 \( 1 + (0.937 - 0.347i)T \)
67 \( 1 + (-0.576 - 0.816i)T \)
71 \( 1 + (-0.282 - 0.959i)T \)
73 \( 1 + (-0.0954 + 0.995i)T \)
79 \( 1 + (-0.881 - 0.472i)T \)
83 \( 1 + (-0.149 - 0.988i)T \)
89 \( 1 + (-0.0682 + 0.997i)T \)
97 \( 1 + (0.410 - 0.911i)T \)
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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

−23.75129987789981732489038321010, −22.88093713105967831196604635505, −21.95551726169628165889745733000, −20.99700890897909681173245912613, −20.31442591404557612367125658644, −19.06935803678953688897262294175, −18.301735635051098689539258844282, −17.874492014558858890762122311146, −16.32340760492749355682937538577, −15.802567094327409178782077943903, −14.67164515970708856482108436511, −14.363778461963183443036996069159, −13.67841764344062279538612615458, −12.54076544502039688899136457297, −11.45997873376553973884923016614, −10.33508775383143195123824858060, −9.156903375056446126271454552674, −8.26323173455918380097905875673, −7.55458104711005276136973449226, −6.97574700785367066054371798177, −5.71071982246251652023004882119, −4.38860274260017768326630494940, −3.6789212861472961915328911337, −2.675018656499890540661365477, −0.995592125612518896743573462819, 1.10163239368337096320532415935, 1.69876208415586742964723595251, 3.1464023543524275729436509411, 3.89907866520478351329167658698, 4.82772233153065483506244369639, 5.74060118335238830523421808457, 7.79254607895484768632967283064, 8.25494673618732753120405205717, 9.147861287674247833572858968217, 9.97553970243235697904197940985, 11.155328119969031250558494849810, 11.934147167953606618139936164313, 12.80026513331491558870433963175, 13.66062365592846076724948025655, 14.38521176936113613182541438829, 15.30220500610571687893290327659, 16.089898830552738831507496200496, 17.49693238780128994850252583947, 18.42304436304961038463639843010, 19.15633245850073645273912363521, 20.04483707531958634777252860606, 20.64697293965374337294573892389, 21.16158081240551401146835874652, 21.95678926308441553462801488574, 23.34148462340823979761901594243

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