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Properties

Label 2-85-85.4-c1-0-7
Degree $2$
Conductor $85$
Sign $-0.980 - 0.197i$
Analytic cond. $0.678728$
Root an. cond. $0.823849$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (−1 − i)3-s − 4-s + (−2 + i)5-s + (1 + i)6-s + (−3 + 3i)7-s + 3·8-s − i·9-s + (2 − i)10-s + (−3 − 3i)11-s + (1 + i)12-s + (3 − 3i)14-s + (3 + i)15-s − 16-s + (−1 − 4i)17-s + i·18-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.577 − 0.577i)3-s − 0.5·4-s + (−0.894 + 0.447i)5-s + (0.408 + 0.408i)6-s + (−1.13 + 1.13i)7-s + 1.06·8-s − 0.333i·9-s + (0.632 − 0.316i)10-s + (−0.904 − 0.904i)11-s + (0.288 + 0.288i)12-s + (0.801 − 0.801i)14-s + (0.774 + 0.258i)15-s − 0.250·16-s + (−0.242 − 0.970i)17-s + 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.980 - 0.197i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ -0.980 - 0.197i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2 - i)T \)
17 \( 1 + (1 + 4i)T \)
good2 \( 1 + T + 2T^{2} \)
3 \( 1 + (1 + i)T + 3iT^{2} \)
7 \( 1 + (3 - 3i)T - 7iT^{2} \)
11 \( 1 + (3 + 3i)T + 11iT^{2} \)
13 \( 1 - 13T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + (-1 + i)T - 23iT^{2} \)
29 \( 1 + (3 - 3i)T - 29iT^{2} \)
31 \( 1 + (1 - i)T - 31iT^{2} \)
37 \( 1 + (-3 - 3i)T + 37iT^{2} \)
41 \( 1 + (3 + 3i)T + 41iT^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 + 2iT - 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + (-1 - i)T + 61iT^{2} \)
67 \( 1 + 6iT - 67T^{2} \)
71 \( 1 + (-3 + 3i)T - 71iT^{2} \)
73 \( 1 + (-3 - 3i)T + 73iT^{2} \)
79 \( 1 + (7 + 7i)T + 79iT^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (-3 - 3i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.42611989823108419734907230713, −12.48684728758773046959987897227, −11.59762270034917081739924900375, −10.32552968461798858128312043103, −9.119068847231446390664997108869, −8.085648823994533951087094693706, −6.79747013306469590511452414662, −5.50118783410996279208378411556, −3.32137479342394424359904830677, 0, 4.01696982023488393293456825090, 4.95977831271023141004118810141, 7.09993871166123286783735789360, 8.097617687612107227145122256818, 9.536000449672005898168666201275, 10.32599067382036335971290078469, 11.18274439729825220667769445726, 12.94830783645454108530989986535, 13.32630533814246610775881387999

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