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Properties

Label 2-85-17.9-c1-0-3
Degree $2$
Conductor $85$
Sign $0.294 - 0.955i$
Analytic cond. $0.678728$
Root an. cond. $0.823849$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.09 + 1.09i)2-s + (2.77 + 1.15i)3-s − 0.419i·4-s + (0.382 − 0.923i)5-s + (−4.32 + 1.79i)6-s + (−1.32 − 3.19i)7-s + (−1.73 − 1.73i)8-s + (4.27 + 4.27i)9-s + (0.595 + 1.43i)10-s + (−3.92 + 1.62i)11-s + (0.483 − 1.16i)12-s − 0.127i·13-s + (4.96 + 2.05i)14-s + (2.12 − 2.12i)15-s + 4.66·16-s + (−4.11 + 0.193i)17-s + ⋯
L(s)  = 1  + (−0.777 + 0.777i)2-s + (1.60 + 0.664i)3-s − 0.209i·4-s + (0.171 − 0.413i)5-s + (−1.76 + 0.731i)6-s + (−0.499 − 1.20i)7-s + (−0.614 − 0.614i)8-s + (1.42 + 1.42i)9-s + (0.188 + 0.454i)10-s + (−1.18 + 0.489i)11-s + (0.139 − 0.336i)12-s − 0.0353i·13-s + (1.32 + 0.549i)14-s + (0.549 − 0.549i)15-s + 1.16·16-s + (−0.998 + 0.0468i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.294 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.294 - 0.955i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ 0.294 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.784527 + 0.578962i\)
\(L(\frac12)\) \(\approx\) \(0.784527 + 0.578962i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.382 + 0.923i)T \)
17 \( 1 + (4.11 - 0.193i)T \)
good2 \( 1 + (1.09 - 1.09i)T - 2iT^{2} \)
3 \( 1 + (-2.77 - 1.15i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (1.32 + 3.19i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (3.92 - 1.62i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + 0.127iT - 13T^{2} \)
19 \( 1 + (-1.81 + 1.81i)T - 19iT^{2} \)
23 \( 1 + (-3.00 + 1.24i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (1.87 - 4.53i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-4.95 - 2.05i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (1.63 + 0.677i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-3.85 - 9.29i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-1.79 - 1.79i)T + 43iT^{2} \)
47 \( 1 + 4.59iT - 47T^{2} \)
53 \( 1 + (-1.15 + 1.15i)T - 53iT^{2} \)
59 \( 1 + (4.34 + 4.34i)T + 59iT^{2} \)
61 \( 1 + (1.54 + 3.73i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 6.88T + 67T^{2} \)
71 \( 1 + (6.66 + 2.76i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (5.59 - 13.5i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (4.75 - 1.97i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-10.2 + 10.2i)T - 83iT^{2} \)
89 \( 1 - 0.600iT - 89T^{2} \)
97 \( 1 + (2.93 - 7.09i)T + (-68.5 - 68.5i)T^{2} \)
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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

−14.73165090128531663686775856847, −13.46064690164771067306134289972, −12.94252736877124134952363595356, −10.51948075369233928363166974878, −9.684679158294848214187703316714, −8.817581727842394682961901378373, −7.84991687850485776524214418464, −6.98046322637125613878017282889, −4.53968901180004964880402790617, −3.06045011009649777364574448665, 2.26527285271161702860616254555, 2.97881761641922208990511247548, 5.94717550848239506590141142360, 7.65198205947563186366460164248, 8.745910606947320553786258201445, 9.330462709402958446569956007474, 10.50179204478189145769082746530, 11.89851705744823061792927714786, 13.03394027331866263843404685158, 13.89387786816538677169346709168

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