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Properties

Label 1-2015-2015.278-r1-0-0
Degree $1$
Conductor $2015$
Sign $-0.966 + 0.256i$
Analytic cond. $216.541$
Root an. cond. $216.541$
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  − 2-s − i·3-s + 4-s + i·6-s + 7-s − 8-s − 9-s − i·11-s − i·12-s − 14-s + 16-s − i·17-s + 18-s + i·19-s − i·21-s + i·22-s + ⋯
L(s)  = 1  − 2-s − i·3-s + 4-s + i·6-s + 7-s − 8-s − 9-s − i·11-s − i·12-s − 14-s + 16-s − i·17-s + 18-s + i·19-s − i·21-s + i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.966 + 0.256i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (278, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ -0.966 + 0.256i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09823231911 - 0.7526215571i\)
\(L(\frac12)\) \(\approx\) \(-0.09823231911 - 0.7526215571i\)
\(L(1)\) \(\approx\) \(0.6404898528 - 0.3149153619i\)
\(L(1)\) \(\approx\) \(0.6404898528 - 0.3149153619i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
31 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
7 \( 1 - iT \)
11 \( 1 + T \)
17 \( 1 + iT \)
19 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 - T \)
37 \( 1 - iT \)
41 \( 1 - iT \)
43 \( 1 \)
47 \( 1 - T \)
53 \( 1 \)
59 \( 1 + T \)
61 \( 1 - iT \)
67 \( 1 + T \)
71 \( 1 + iT \)
73 \( 1 \)
79 \( 1 - iT \)
83 \( 1 + iT \)
89 \( 1 + iT \)
97 \( 1 + iT \)
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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

−20.11707908950842477901231729646, −19.621132059619072957302628570473, −18.558487738987783062845485311599, −17.759444674654447216593021873323, −17.286290559492207159470659209459, −16.73048577604142894249094759033, −15.70765356624210519129647927394, −15.20030351629834196752540639980, −14.70388621640614170255475693056, −13.79320300704889171594398942359, −12.379877375768343362229755530548, −11.88310756381826069116262603928, −10.81947522765872013100085868593, −10.58762799626549903320940627447, −9.77171033408918114964165332776, −8.84407159249784691327350957131, −8.46332627879337258753412268329, −7.54079805625848391441797591335, −6.673965832842724242410921259036, −5.70577185775852427948910178371, −4.76718855360922817588872055068, −4.122864381563187373494342378480, −2.85946067314002410453107494072, −2.12671317513433828275411819939, −1.07047345876608679014679453184, 0.20449301770395990629780784380, 1.10384241793554808500867596787, 1.761166512702712581720923297098, 2.70178392703173576161532239826, 3.56012969520448403316452588845, 5.15962623857489527848617147733, 5.827117604270268823976999412036, 6.69363361927677714320204767587, 7.51162472975905281688633140331, 8.06846701912846284351112582123, 8.66383152333514373724642425696, 9.47004386480423348221286570488, 10.605580688406997990423023949905, 11.19960431548505510561241923739, 11.8892787514837144021365931364, 12.38272114451372812667767716648, 13.748613009824104569304331863276, 14.04340394386146400905992116399, 15.01716808513564342561408654126, 15.940478925942448422736521045154, 16.67538363280979737014135745081, 17.41469019928053067324365062731, 17.98262138319615532967831963406, 18.556244361352748063061329002633, 19.21963553263679774046987476894

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