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Properties

Label 2-170-85.3-c1-0-8
Degree $2$
Conductor $170$
Sign $-0.980 + 0.195i$
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  + (0.382 − 0.923i)2-s + (−1.65 − 2.47i)3-s + (−0.707 − 0.707i)4-s + (0.488 − 2.18i)5-s + (−2.92 + 0.581i)6-s + (0.420 + 2.11i)7-s + (−0.923 + 0.382i)8-s + (−2.25 + 5.44i)9-s + (−1.82 − 1.28i)10-s + (0.581 + 2.92i)11-s + (−0.581 + 2.92i)12-s − 5.10i·13-s + (2.11 + 0.420i)14-s + (−6.21 + 2.40i)15-s + i·16-s + (2.91 − 2.91i)17-s + ⋯
L(s)  = 1  + (0.270 − 0.653i)2-s + (−0.956 − 1.43i)3-s + (−0.353 − 0.353i)4-s + (0.218 − 0.975i)5-s + (−1.19 + 0.237i)6-s + (0.159 + 0.799i)7-s + (−0.326 + 0.135i)8-s + (−0.751 + 1.81i)9-s + (−0.578 − 0.406i)10-s + (0.175 + 0.880i)11-s + (−0.167 + 0.844i)12-s − 1.41i·13-s + (0.565 + 0.112i)14-s + (−1.60 + 0.620i)15-s + 0.250i·16-s + (0.708 − 0.706i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.980 + 0.195i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.980 + 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0900196 - 0.913436i\)
\(L(\frac12)\) \(\approx\) \(0.0900196 - 0.913436i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 + 0.923i)T \)
5 \( 1 + (-0.488 + 2.18i)T \)
17 \( 1 + (-2.91 + 2.91i)T \)
good3 \( 1 + (1.65 + 2.47i)T + (-1.14 + 2.77i)T^{2} \)
7 \( 1 + (-0.420 - 2.11i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (-0.581 - 2.92i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + 5.10iT - 13T^{2} \)
19 \( 1 + (2.47 + 5.97i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-1.26 - 0.847i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (-3.82 - 5.71i)T + (-11.0 + 26.7i)T^{2} \)
31 \( 1 + (0.942 - 4.73i)T + (-28.6 - 11.8i)T^{2} \)
37 \( 1 + (-6.05 + 4.04i)T + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (-2.01 + 3.01i)T + (-15.6 - 37.8i)T^{2} \)
43 \( 1 + (0.718 + 1.73i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 4.65T + 47T^{2} \)
53 \( 1 + (-2.70 - 1.11i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-3.66 - 1.51i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (3.45 + 2.30i)T + (23.3 + 56.3i)T^{2} \)
67 \( 1 + (-2.25 - 2.25i)T + 67iT^{2} \)
71 \( 1 + (-5.63 - 1.12i)T + (65.5 + 27.1i)T^{2} \)
73 \( 1 + (1.91 - 9.62i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (-15.1 + 3.01i)T + (72.9 - 30.2i)T^{2} \)
83 \( 1 + (3.74 - 9.03i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-5.41 - 5.41i)T + 89iT^{2} \)
97 \( 1 + (0.523 - 2.63i)T + (-89.6 - 37.1i)T^{2} \)
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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

−12.51774911996521863370462788541, −11.72100375173078378401178129369, −10.65701176864639788450820460754, −9.280336456083545671282514823657, −8.142927370540076693375727739752, −6.91195526883456638827117075693, −5.53834121607351074619121160526, −5.01280645572645696758202376049, −2.43824971132319557885904878851, −0.951514792641161660305703175231, 3.65344741876581363747780005158, 4.39047401697284590398169975868, 5.93193103315388870979854295032, 6.45128334360327751704291046295, 8.007863680400907246286708496778, 9.516153584145098225948290836808, 10.29251145089180400171880601835, 11.12746863770293035333417368620, 11.90941854112076407612091815921, 13.59116091355635087116090216205

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