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Properties

Label 1-4017-4017.1247-r0-0-0
Degree $1$
Conductor $4017$
Sign $0.949 - 0.313i$
Analytic cond. $18.6548$
Root an. cond. $18.6548$
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  + (−0.389 + 0.920i)2-s + (−0.696 − 0.717i)4-s + (−0.816 − 0.577i)5-s + (0.779 − 0.626i)7-s + (0.932 − 0.361i)8-s + (0.850 − 0.526i)10-s + (−0.992 + 0.122i)11-s + (0.273 + 0.961i)14-s + (−0.0307 + 0.999i)16-s + (−0.650 + 0.759i)17-s + (−0.552 + 0.833i)19-s + (0.153 + 0.988i)20-s + (0.273 − 0.961i)22-s + (0.602 − 0.798i)23-s + (0.332 + 0.943i)25-s + ⋯
L(s)  = 1  + (−0.389 + 0.920i)2-s + (−0.696 − 0.717i)4-s + (−0.816 − 0.577i)5-s + (0.779 − 0.626i)7-s + (0.932 − 0.361i)8-s + (0.850 − 0.526i)10-s + (−0.992 + 0.122i)11-s + (0.273 + 0.961i)14-s + (−0.0307 + 0.999i)16-s + (−0.650 + 0.759i)17-s + (−0.552 + 0.833i)19-s + (0.153 + 0.988i)20-s + (0.273 − 0.961i)22-s + (0.602 − 0.798i)23-s + (0.332 + 0.943i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4017\)    =    \(3 \cdot 13 \cdot 103\)
Sign: $0.949 - 0.313i$
Analytic conductor: \(18.6548\)
Root analytic conductor: \(18.6548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4017} (1247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4017,\ (0:\ ),\ 0.949 - 0.313i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8391821836 - 0.1351282215i\)
\(L(\frac12)\) \(\approx\) \(0.8391821836 - 0.1351282215i\)
\(L(1)\) \(\approx\) \(0.6930823979 + 0.1289336269i\)
\(L(1)\) \(\approx\) \(0.6930823979 + 0.1289336269i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.389 + 0.920i)T \)
5 \( 1 + (-0.816 - 0.577i)T \)
7 \( 1 + (0.779 - 0.626i)T \)
11 \( 1 + (-0.992 + 0.122i)T \)
17 \( 1 + (-0.650 + 0.759i)T \)
19 \( 1 + (-0.552 + 0.833i)T \)
23 \( 1 + (0.602 - 0.798i)T \)
29 \( 1 + (0.908 - 0.417i)T \)
31 \( 1 + (-0.850 - 0.526i)T \)
37 \( 1 + (0.739 + 0.673i)T \)
41 \( 1 + (0.816 - 0.577i)T \)
43 \( 1 + (0.952 + 0.303i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.998 - 0.0615i)T \)
59 \( 1 + (-0.779 - 0.626i)T \)
61 \( 1 + (-0.982 - 0.183i)T \)
67 \( 1 + (-0.153 + 0.988i)T \)
71 \( 1 + (0.908 + 0.417i)T \)
73 \( 1 + (0.0922 + 0.995i)T \)
79 \( 1 + (0.0922 - 0.995i)T \)
83 \( 1 + (-0.153 - 0.988i)T \)
89 \( 1 + (0.273 + 0.961i)T \)
97 \( 1 + (-0.650 - 0.759i)T \)
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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

−18.386871692582924914683039423717, −18.08765416376983786124740342260, −17.55109845729947320797173977521, −16.48953094514507411512646958355, −15.68908320937264744711834817658, −15.26218097900874156902734917778, −14.289282254447129431033205803095, −13.71734511966780691715739961590, −12.73009775804353459971519138821, −12.312653745937754175943427543235, −11.37099795368541144797719815038, −10.95018628072178242191236578454, −10.65894342653108662127951363436, −9.32232977121191142410231114025, −8.99273246511019703369280173088, −7.99566655236395593992989205566, −7.65082178567027164763638198751, −6.81640268380337929617599379497, −5.5848959942479545450935292230, −4.72093954375323695328059691913, −4.29773279140842146479174608755, −2.99212253866680396337506879897, −2.78578819311425505261543096291, −1.864980589144289256132036702618, −0.70066394251556567008708127786, 0.43077462293937968891981185692, 1.3247409339735338797023619257, 2.3276100737522635070964503266, 3.77132547249921366272769664179, 4.41918125177220785052614106197, 4.87175482448414202120448926345, 5.768972748498632788354455320232, 6.58515900654195410466047031550, 7.5215717941004615844957812156, 7.9146792446443235979846921174, 8.466865915418252987919233526198, 9.14133483920400922763326811703, 10.22266470164568117040413974052, 10.727544411673797320851373582861, 11.35191115677737387592704669989, 12.57272146604251274425080970552, 12.942172317513002749885468925584, 13.81167459883879458520566181223, 14.599693071878222112816679042752, 15.12707760843705217941689865259, 15.76648033662340661146494501640, 16.4185780304163791063058823353, 17.08861821449413489916274981815, 17.513881941408132594985423229210, 18.452887728539871879003518883158

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