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Properties

Label 2-170-85.64-c1-0-1
Degree $2$
Conductor $170$
Sign $-0.197 - 0.980i$
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 + (−1.70 + 1.70i)3-s + 4-s + (2.12 − 0.707i)5-s + (1.70 − 1.70i)6-s + (1 + i)7-s − 8-s − 2.82i·9-s + (−2.12 + 0.707i)10-s + (−4.41 + 4.41i)11-s + (−1.70 + 1.70i)12-s + 3i·13-s + (−1 − i)14-s + (−2.41 + 4.82i)15-s + 16-s + (2.12 + 3.53i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.985 + 0.985i)3-s + 0.5·4-s + (0.948 − 0.316i)5-s + (0.696 − 0.696i)6-s + (0.377 + 0.377i)7-s − 0.353·8-s − 0.942i·9-s + (−0.670 + 0.223i)10-s + (−1.33 + 1.33i)11-s + (−0.492 + 0.492i)12-s + 0.832i·13-s + (−0.267 − 0.267i)14-s + (−0.623 + 1.24i)15-s + 0.250·16-s + (0.514 + 0.857i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.197 - 0.980i$
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: \((2,\ 170,\ (\ :1/2),\ -0.197 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.413486 + 0.505333i\)
\(L(\frac12)\) \(\approx\) \(0.413486 + 0.505333i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 + 0.707i)T \)
17 \( 1 + (-2.12 - 3.53i)T \)
good3 \( 1 + (1.70 - 1.70i)T - 3iT^{2} \)
7 \( 1 + (-1 - i)T + 7iT^{2} \)
11 \( 1 + (4.41 - 4.41i)T - 11iT^{2} \)
13 \( 1 - 3iT - 13T^{2} \)
19 \( 1 + 1.24iT - 19T^{2} \)
23 \( 1 + (2.82 + 2.82i)T + 23iT^{2} \)
29 \( 1 + (0.707 + 0.707i)T + 29iT^{2} \)
31 \( 1 + (-7.36 - 7.36i)T + 31iT^{2} \)
37 \( 1 + (3.24 - 3.24i)T - 37iT^{2} \)
41 \( 1 + (-1.58 + 1.58i)T - 41iT^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 + 4.41iT - 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + 6.89iT - 59T^{2} \)
61 \( 1 + (-1.87 + 1.87i)T - 61iT^{2} \)
67 \( 1 + 2.48iT - 67T^{2} \)
71 \( 1 + (-2.29 - 2.29i)T + 71iT^{2} \)
73 \( 1 + (-4.36 + 4.36i)T - 73iT^{2} \)
79 \( 1 + (-8.24 + 8.24i)T - 79iT^{2} \)
83 \( 1 + 4.24T + 83T^{2} \)
89 \( 1 + 5.48T + 89T^{2} \)
97 \( 1 + (4.12 - 4.12i)T - 97iT^{2} \)
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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

−12.69437505848002432373417163719, −11.92140614873080879192651680771, −10.59458234543289520692774973808, −10.21999242831252824385014062502, −9.331093417074770181909538288357, −8.120643014767994056446243577361, −6.57809881474586779126552393297, −5.42362586164330365067503332566, −4.61396679101289081291319229855, −2.12881836399994994960256539840, 0.876178125641113829981671910039, 2.69736508549986738926955025885, 5.53805942215431286644798672492, 5.98718256841431981560261418285, 7.37616267337189092173334361843, 8.072766323650021338088375161053, 9.658300295647341822290386271345, 10.69713591531437631803756256432, 11.23131380918739889531855583733, 12.43208153142033867245532302145

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