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Properties

Label 2-85-85.4-c1-0-3
Degree $2$
Conductor $85$
Sign $0.902 - 0.429i$
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  + 2-s + (1 + i)3-s − 4-s + (−1 + 2i)5-s + (1 + i)6-s + (3 − 3i)7-s − 3·8-s − i·9-s + (−1 + 2i)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 + (−0.447 + 0.894i)5-s + (0.408 + 0.408i)6-s + (1.13 − 1.13i)7-s − 1.06·8-s − 0.333i·9-s + (−0.316 + 0.632i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.902 - 0.429i$
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: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ 0.902 - 0.429i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27685 + 0.288373i\)
\(L(\frac12)\) \(\approx\) \(1.27685 + 0.288373i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1 - 2i)T \)
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

−14.40160283512726406922595361117, −13.70071605705661377357561342452, −12.31514992730239331153242712356, −10.93744248030338334125964702834, −10.19238182224511771261920467036, −8.568972956638657282222548941950, −7.65144474197002644281810748719, −5.79663938906687122063619373336, −4.16080075191035978763604078265, −3.43197126874855401031714599883, 2.44264460611605416164578916134, 4.72497226585874361076547094617, 5.27091516089608358425373647377, 7.58763626479385463945349547505, 8.462854333626999577868475916988, 9.355977995757028984368500482249, 11.41117665555159837062054382400, 12.38290871747372188836599527133, 13.10032362330901460964222778441, 14.01362185971356011139195805864

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