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Properties

Label 1-85-85.19-r0-0-0
Degree $1$
Conductor $85$
Sign $0.998 - 0.0465i$
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  − i·2-s + (−0.707 + 0.707i)3-s − 4-s + (0.707 + 0.707i)6-s + (0.707 + 0.707i)7-s + i·8-s − i·9-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)12-s + 13-s + (0.707 − 0.707i)14-s + 16-s − 18-s + i·19-s − 21-s + (0.707 − 0.707i)22-s + ⋯
L(s)  = 1  − i·2-s + (−0.707 + 0.707i)3-s − 4-s + (0.707 + 0.707i)6-s + (0.707 + 0.707i)7-s + i·8-s − i·9-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)12-s + 13-s + (0.707 − 0.707i)14-s + 16-s − 18-s + i·19-s − 21-s + (0.707 − 0.707i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.998 - 0.0465i$
Analytic conductor: \(0.394738\)
Root analytic conductor: \(0.394738\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (0:\ ),\ 0.998 - 0.0465i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7775395004 + 0.01809283341i\)
\(L(\frac12)\) \(\approx\) \(0.7775395004 + 0.01809283341i\)
\(L(1)\) \(\approx\) \(0.8407159625 - 0.1050566136i\)
\(L(1)\) \(\approx\) \(0.8407159625 - 0.1050566136i\)

Euler product

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

−30.53507000185405041772125130829, −30.04858652955149855529112422826, −28.40589574738200141221905271823, −27.56312686380187118855308073674, −26.473644101911759988819496707601, −25.2211351793321170462088530610, −24.2005304380223869037735019668, −23.61304009103839021206073351956, −22.59228914363120954267022150546, −21.46222163710098929258200895237, −19.64523901179275460983969577536, −18.45222146304630693815127174713, −17.51241137632729609830841354031, −16.769176484172081473175311476846, −15.58935368122872438274466394098, −13.943859419391369796393901048135, −13.462438427169651541238197271991, −11.81135283739595836072941789783, −10.63855544779075700041224019029, −8.78765586723070371647110147652, −7.64427169775186883399198061220, −6.57019575111955957927175845129, −5.475474102685949374133547167123, −4.04627465599664789669287851392, −1.134652007819920911760003131181, 1.67927659016608564269863338405, 3.6677952290932752371909661607, 4.79182024700133175772334714784, 6.05534207719123430907242771203, 8.39293142783488076211101406842, 9.52460401111150640628797657747, 10.6618034936228733247131837963, 11.694205830409226200875103559862, 12.44485242411014734965186582801, 14.195437686651900585310516221403, 15.25815329403275679457104087982, 16.788713054212510931338540569858, 17.886010085837508345980907830237, 18.69036358481095088483680263579, 20.40598830780810919024042826519, 20.98559999708818753705381744941, 22.15156846573392504114188825723, 22.81308236109335382636785807307, 24.050912948477063457433057987833, 25.70786444088797047730374001223, 27.02660801806620604199884333954, 27.92882257634774215072990284045, 28.32585639624382784325119773510, 29.62142154427089525653008381700, 30.656654015818422915248216289230

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