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Properties

Label 1-85-85.22-r0-0-0
Degree $1$
Conductor $85$
Sign $0.922 - 0.386i$
Analytic cond. $0.394738$
Root an. cond. $0.394738$
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  + (−0.707 − 0.707i)2-s + (0.923 + 0.382i)3-s + i·4-s + (−0.382 − 0.923i)6-s + (−0.382 − 0.923i)7-s + (0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (0.382 + 0.923i)11-s + (−0.382 + 0.923i)12-s + 13-s + (−0.382 + 0.923i)14-s − 16-s − i·18-s + (0.707 − 0.707i)19-s − i·21-s + (0.382 − 0.923i)22-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (0.923 + 0.382i)3-s + i·4-s + (−0.382 − 0.923i)6-s + (−0.382 − 0.923i)7-s + (0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (0.382 + 0.923i)11-s + (−0.382 + 0.923i)12-s + 13-s + (−0.382 + 0.923i)14-s − 16-s − i·18-s + (0.707 − 0.707i)19-s − i·21-s + (0.382 − 0.923i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.922 - 0.386i$
Analytic conductor: \(0.394738\)
Root analytic conductor: \(0.394738\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (0:\ ),\ 0.922 - 0.386i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9153440286 - 0.1838165876i\)
\(L(\frac12)\) \(\approx\) \(0.9153440286 - 0.1838165876i\)
\(L(1)\) \(\approx\) \(0.9629729231 - 0.1667101039i\)
\(L(1)\) \(\approx\) \(0.9629729231 - 0.1667101039i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (0.923 + 0.382i)T \)
7 \( 1 + (-0.382 - 0.923i)T \)
11 \( 1 + (0.382 + 0.923i)T \)
13 \( 1 + T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (-0.923 + 0.382i)T \)
29 \( 1 + (-0.923 - 0.382i)T \)
31 \( 1 + (0.382 - 0.923i)T \)
37 \( 1 + (-0.923 - 0.382i)T \)
41 \( 1 + (-0.923 + 0.382i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 - T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (0.923 - 0.382i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.382 + 0.923i)T \)
73 \( 1 + (-0.382 + 0.923i)T \)
79 \( 1 + (-0.382 - 0.923i)T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.382 + 0.923i)T \)
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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

−31.02270612493989696073537259282, −29.659874032527615353840679648497, −28.57629545318949943937993959361, −27.43145103862959100583319081446, −26.36871693050308580090879617432, −25.51365523857429334595703506864, −24.71666896192574438406384021808, −23.87218516935757756989925194731, −22.39164009731681268498756881142, −20.88136723636168541699612804860, −19.688716625135326199027415533566, −18.72073543975814796777023019145, −18.15649067663381311789609013316, −16.40398747024723385515677199199, −15.56861997693161397342861815166, −14.40105141544392212802098517723, −13.459502349675083138737016394737, −11.834609752335949660615392483294, −10.11731908490325898745576309317, −8.85468922881508184730955885268, −8.30570068255287426157807671635, −6.76917640314439923140342171327, −5.69419081218228498788741865719, −3.42053129634810597613967514474, −1.63821709722534523897851063561, 1.691482926362417444885704236761, 3.33643157289803732815401311268, 4.291399181366467096049173106058, 7.0084981767532850879113275372, 8.072908086531523831979301224615, 9.40633484419589703967285901171, 10.1143483925876720813396415097, 11.405703658796288540761325182, 13.0225402860067333200088001937, 13.83834688551077615783614421403, 15.48508552928735558890126003430, 16.55188619366019991226596211392, 17.802871817731663455393288912631, 19.03347652624144040646956144635, 20.14809787698167031356364142734, 20.48507812825529800837355366159, 21.8168896187675971020326870154, 22.94879109904264248986169386884, 24.69443015540158179839122647923, 26.072198900305498811638704289346, 26.17013534316152355455189828806, 27.589299056977549218198934356702, 28.33808722609564366757171051358, 29.87211484345893164568997686294, 30.46086225467897291674456773004

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