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Properties

Label 2-170-85.64-c1-0-2
Degree $2$
Conductor $170$
Sign $0.429 - 0.902i$
Analytic cond. $1.35745$
Root an. cond. $1.16509$
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 + (−2 + i)5-s + (−1 + i)6-s + (3 + 3i)7-s + 8-s + i·9-s + (−2 + i)10-s + (1 − i)11-s + (−1 + i)12-s − 4i·13-s + (3 + 3i)14-s + (1 − 3i)15-s + 16-s + (−1 + 4i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.577 + 0.577i)3-s + 0.5·4-s + (−0.894 + 0.447i)5-s + (−0.408 + 0.408i)6-s + (1.13 + 1.13i)7-s + 0.353·8-s + 0.333i·9-s + (−0.632 + 0.316i)10-s + (0.301 − 0.301i)11-s + (−0.288 + 0.288i)12-s − 1.10i·13-s + (0.801 + 0.801i)14-s + (0.258 − 0.774i)15-s + 0.250·16-s + (−0.242 + 0.970i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.429 - 0.902i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.429 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16423 + 0.735243i\)
\(L(\frac12)\) \(\approx\) \(1.16423 + 0.735243i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + (2 - i)T \)
17 \( 1 + (1 - 4i)T \)
good3 \( 1 + (1 - i)T - 3iT^{2} \)
7 \( 1 + (-3 - 3i)T + 7iT^{2} \)
11 \( 1 + (-1 + i)T - 11iT^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + (5 + 5i)T + 23iT^{2} \)
29 \( 1 + (-7 - 7i)T + 29iT^{2} \)
31 \( 1 + (-1 - i)T + 31iT^{2} \)
37 \( 1 + (-5 + 5i)T - 37iT^{2} \)
41 \( 1 + (-1 + i)T - 41iT^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 2iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 + (9 - 9i)T - 61iT^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 + (-1 - i)T + 71iT^{2} \)
73 \( 1 + (1 - i)T - 73iT^{2} \)
79 \( 1 + (-3 + 3i)T - 79iT^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (5 - 5i)T - 97iT^{2} \)
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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

−12.66834478168736826778636778428, −11.90744628447711914546163159567, −10.98984956864593493188549839051, −10.58879304575573474855103256916, −8.637831267226108235874119331219, −7.84169226384929036796104729059, −6.28219970826725314459548453440, −5.17416870779593159403904206997, −4.31746307178271711054434627085, −2.65378181486934291426092448968, 1.36130075350983249833438177699, 3.96435464936665340622172962678, 4.65670264009737724695989235879, 6.21918428108980912645740534679, 7.33478917357118335549553531501, 8.016811289869478062046925192278, 9.706844422451958685104898079194, 11.21623606705598553803781852991, 11.74212759132257079134006454488, 12.30872379070584069272965824626

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