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Properties

Label 2-85-5.4-c1-0-0
Degree $2$
Conductor $85$
Sign $-0.717 - 0.696i$
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  + 2.31i·2-s − 0.203i·3-s − 3.36·4-s + (−1.55 + 1.60i)5-s + 0.470·6-s + 0.683i·7-s − 3.16i·8-s + 2.95·9-s + (−3.71 − 3.60i)10-s + 3.68·11-s + 0.683i·12-s − 4.43i·13-s − 1.58·14-s + (0.326 + 0.316i)15-s + 0.593·16-s − i·17-s + ⋯
L(s)  = 1  + 1.63i·2-s − 0.117i·3-s − 1.68·4-s + (−0.696 + 0.717i)5-s + 0.192·6-s + 0.258i·7-s − 1.11i·8-s + 0.986·9-s + (−1.17 − 1.14i)10-s + 1.10·11-s + 0.197i·12-s − 1.23i·13-s − 0.423·14-s + (0.0842 + 0.0816i)15-s + 0.148·16-s − 0.242i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.717 - 0.696i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ -0.717 - 0.696i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.331436 + 0.817960i\)
\(L(\frac12)\) \(\approx\) \(0.331436 + 0.817960i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.55 - 1.60i)T \)
17 \( 1 + iT \)
good2 \( 1 - 2.31iT - 2T^{2} \)
3 \( 1 + 0.203iT - 3T^{2} \)
7 \( 1 - 0.683iT - 7T^{2} \)
11 \( 1 - 3.68T + 11T^{2} \)
13 \( 1 + 4.43iT - 13T^{2} \)
19 \( 1 - 1.03T + 19T^{2} \)
23 \( 1 - 4.52iT - 23T^{2} \)
29 \( 1 - 3.69T + 29T^{2} \)
31 \( 1 + 10.8T + 31T^{2} \)
37 \( 1 - 0.308iT - 37T^{2} \)
41 \( 1 + 6.15T + 41T^{2} \)
43 \( 1 + 7.88iT - 43T^{2} \)
47 \( 1 - 4.43iT - 47T^{2} \)
53 \( 1 + 11.4iT - 53T^{2} \)
59 \( 1 - 2T + 59T^{2} \)
61 \( 1 - 9.94T + 61T^{2} \)
67 \( 1 - 9.16iT - 67T^{2} \)
71 \( 1 + 9.37T + 71T^{2} \)
73 \( 1 - 2.26iT - 73T^{2} \)
79 \( 1 + 7.42T + 79T^{2} \)
83 \( 1 - 8.92iT - 83T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 + 12.5iT - 97T^{2} \)
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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

−14.96289471798943453735081011909, −14.03547804584369859740268541024, −12.81459608668323257132852710297, −11.51106472295164535490443351605, −9.998637889505309153297139003273, −8.648457557085089159805181634716, −7.46466157671337169734102840986, −6.84753743345032280555999676393, −5.47425074251320789156602539935, −3.84281891265127316846266739828, 1.48594716639625440615454634913, 3.82968295194503621722075766395, 4.52232654991690591442318789920, 7.00037624176233505161155877050, 8.786742033720323352214993238608, 9.563633682556837462770448084217, 10.77773018970947844274856555189, 11.79166843270580616596967633554, 12.43093412148718433861311082548, 13.40892906567950363194232717018

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