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Properties

Label 1-85-85.58-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: $\chi_{85} (58, \cdot )$
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

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

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