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Properties

Label 2-85-17.8-c1-0-5
Degree $2$
Conductor $85$
Sign $0.483 + 0.875i$
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 + (0.436 − 1.05i)3-s − 0.403i·4-s + (−0.923 − 0.382i)5-s + (−0.676 − 1.63i)6-s + (−3.45 + 1.43i)7-s + (1.74 + 1.74i)8-s + (1.20 + 1.20i)9-s + (−1.43 + 0.593i)10-s + (0.558 + 1.34i)11-s + (−0.425 − 0.176i)12-s − 6.71i·13-s + (−2.21 + 5.35i)14-s + (−0.806 + 0.806i)15-s + 4.64·16-s + (−4.12 + 0.148i)17-s + ⋯
L(s)  = 1  + (0.775 − 0.775i)2-s + (0.251 − 0.608i)3-s − 0.201i·4-s + (−0.413 − 0.171i)5-s + (−0.276 − 0.666i)6-s + (−1.30 + 0.541i)7-s + (0.618 + 0.618i)8-s + (0.400 + 0.400i)9-s + (−0.452 + 0.187i)10-s + (0.168 + 0.406i)11-s + (−0.122 − 0.0508i)12-s − 1.86i·13-s + (−0.593 + 1.43i)14-s + (−0.208 + 0.208i)15-s + 1.16·16-s + (−0.999 + 0.0360i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.483 + 0.875i$
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: \((2,\ 85,\ (\ :1/2),\ 0.483 + 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14191 - 0.673716i\)
\(L(\frac12)\) \(\approx\) \(1.14191 - 0.673716i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.12 - 0.148i)T \)
good2 \( 1 + (-1.09 + 1.09i)T - 2iT^{2} \)
3 \( 1 + (-0.436 + 1.05i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (3.45 - 1.43i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.558 - 1.34i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 6.71iT - 13T^{2} \)
19 \( 1 + (1.32 - 1.32i)T - 19iT^{2} \)
23 \( 1 + (-1.61 - 3.90i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (7.01 + 2.90i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-0.495 + 1.19i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-1.72 + 4.17i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.45 + 0.601i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (4.56 + 4.56i)T + 43iT^{2} \)
47 \( 1 - 2.36iT - 47T^{2} \)
53 \( 1 + (-4.25 + 4.25i)T - 53iT^{2} \)
59 \( 1 + (-7.29 - 7.29i)T + 59iT^{2} \)
61 \( 1 + (-2.90 + 1.20i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 6.08T + 67T^{2} \)
71 \( 1 + (1.46 - 3.53i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-12.9 - 5.35i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (0.916 + 2.21i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (11.9 - 11.9i)T - 83iT^{2} \)
89 \( 1 + 1.59iT - 89T^{2} \)
97 \( 1 + (12.5 + 5.20i)T + (68.5 + 68.5i)T^{2} \)
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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

−13.40732634566187812465668074924, −12.93579585951004146258091740466, −12.39872924621896956282829282995, −11.09504576490670125887790937713, −9.884960758481189962122038831189, −8.309537271834234748970369294953, −7.20134396840418303801662957056, −5.50463038523874269552164680812, −3.80595170404587214488247693041, −2.47672759645299743449605434089, 3.69077418884048571776617518432, 4.51702859598937648737741714164, 6.56214267727134729545756971410, 6.88149721976688501203294746585, 8.995539092348428615786228870534, 9.911455727146389640924969502402, 11.15243178594574564970534817551, 12.71546681158233379190863867139, 13.58329097305387428297619303530, 14.56846912693856927680366548260

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