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Properties

Label 2-85-17.8-c1-0-3
Degree $2$
Conductor $85$
Sign $0.861 + 0.507i$
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  + (−0.528 + 0.528i)2-s + (1.17 − 2.84i)3-s + 1.44i·4-s + (−0.923 − 0.382i)5-s + (0.880 + 2.12i)6-s + (2.98 − 1.23i)7-s + (−1.81 − 1.81i)8-s + (−4.56 − 4.56i)9-s + (0.690 − 0.286i)10-s + (1.04 + 2.52i)11-s + (4.09 + 1.69i)12-s + 4.31i·13-s + (−0.925 + 2.23i)14-s + (−2.17 + 2.17i)15-s − 0.956·16-s + (−3.47 + 2.22i)17-s + ⋯
L(s)  = 1  + (−0.373 + 0.373i)2-s + (0.679 − 1.64i)3-s + 0.720i·4-s + (−0.413 − 0.171i)5-s + (0.359 + 0.867i)6-s + (1.12 − 0.467i)7-s + (−0.643 − 0.643i)8-s + (−1.52 − 1.52i)9-s + (0.218 − 0.0905i)10-s + (0.315 + 0.761i)11-s + (1.18 + 0.489i)12-s + 1.19i·13-s + (−0.247 + 0.596i)14-s + (−0.561 + 0.561i)15-s − 0.239·16-s + (−0.841 + 0.539i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.861 + 0.507i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ 0.861 + 0.507i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.922619 - 0.251503i\)
\(L(\frac12)\) \(\approx\) \(0.922619 - 0.251503i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.923 + 0.382i)T \)
17 \( 1 + (3.47 - 2.22i)T \)
good2 \( 1 + (0.528 - 0.528i)T - 2iT^{2} \)
3 \( 1 + (-1.17 + 2.84i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-2.98 + 1.23i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-1.04 - 2.52i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 4.31iT - 13T^{2} \)
19 \( 1 + (0.897 - 0.897i)T - 19iT^{2} \)
23 \( 1 + (0.188 + 0.454i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (0.410 + 0.170i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (2.11 - 5.10i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-4.09 + 9.88i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (2.00 - 0.830i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-1.52 - 1.52i)T + 43iT^{2} \)
47 \( 1 + 8.39iT - 47T^{2} \)
53 \( 1 + (-1.28 + 1.28i)T - 53iT^{2} \)
59 \( 1 + (2.13 + 2.13i)T + 59iT^{2} \)
61 \( 1 + (-11.2 + 4.67i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 4.21T + 67T^{2} \)
71 \( 1 + (1.48 - 3.59i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (5.97 + 2.47i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (2.76 + 6.67i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-0.160 + 0.160i)T - 83iT^{2} \)
89 \( 1 - 13.3iT - 89T^{2} \)
97 \( 1 + (13.6 + 5.66i)T + (68.5 + 68.5i)T^{2} \)
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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

−14.12384495018135912635278722047, −13.02613305535379477211567049259, −12.21643197598732643501210002393, −11.34577158764887912473525712203, −9.038122943085563692180363056723, −8.277640514884716915290997025452, −7.37143188237714541528540026542, −6.70682957295278224979330017064, −4.09948290286126213997464928233, −1.93543693104800046984715487414, 2.80548896473677538216553205172, 4.52864843745001145845525537944, 5.62142644629249515098599909945, 8.188466659213570999184646802548, 8.912370712636221101703471374563, 9.982645868982822140653082037850, 11.00056935983272965957904987338, 11.45983070103043998185579593999, 13.70637158540265430606187673575, 14.80185163038472687089210222526

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