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Properties

Label 2-170-85.12-c1-0-8
Degree $2$
Conductor $170$
Sign $-0.701 + 0.713i$
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 + (0.567 − 2.85i)3-s + (−0.707 − 0.707i)4-s + (−2.18 − 0.460i)5-s + (−2.41 − 1.61i)6-s + (3.60 + 2.40i)7-s + (−0.923 + 0.382i)8-s + (−5.03 − 2.08i)9-s + (−1.26 + 1.84i)10-s + (−0.113 + 0.170i)11-s + (−2.41 + 1.61i)12-s + 3.32·13-s + (3.60 − 2.40i)14-s + (−2.55 + 5.97i)15-s + i·16-s + (−4.05 − 0.761i)17-s + ⋯
L(s)  = 1  + (0.270 − 0.653i)2-s + (0.327 − 1.64i)3-s + (−0.353 − 0.353i)4-s + (−0.978 − 0.205i)5-s + (−0.986 − 0.659i)6-s + (1.36 + 0.910i)7-s + (−0.326 + 0.135i)8-s + (−1.67 − 0.695i)9-s + (−0.399 + 0.583i)10-s + (−0.0343 + 0.0513i)11-s + (−0.697 + 0.466i)12-s + 0.923·13-s + (0.963 − 0.643i)14-s + (−0.659 + 1.54i)15-s + 0.250i·16-s + (−0.982 − 0.184i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.701 + 0.713i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.701 + 0.713i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.507296 - 1.21028i\)
\(L(\frac12)\) \(\approx\) \(0.507296 - 1.21028i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 + 0.460i)T \)
17 \( 1 + (4.05 + 0.761i)T \)
good3 \( 1 + (-0.567 + 2.85i)T + (-2.77 - 1.14i)T^{2} \)
7 \( 1 + (-3.60 - 2.40i)T + (2.67 + 6.46i)T^{2} \)
11 \( 1 + (0.113 - 0.170i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 - 3.32T + 13T^{2} \)
19 \( 1 + (-1.76 + 0.729i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.769 - 0.153i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-8.95 - 1.78i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 + (4.85 + 7.26i)T + (-11.8 + 28.6i)T^{2} \)
37 \( 1 + (1.88 + 0.374i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (3.42 - 0.682i)T + (37.8 - 15.6i)T^{2} \)
43 \( 1 + (-2.60 - 6.29i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 11.8iT - 47T^{2} \)
53 \( 1 + (-4.78 - 1.97i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (1.58 - 3.81i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-0.209 - 1.05i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (5.49 - 5.49i)T - 67iT^{2} \)
71 \( 1 + (1.33 - 0.892i)T + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (0.677 - 0.452i)T + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (10.3 + 6.88i)T + (30.2 + 72.9i)T^{2} \)
83 \( 1 + (0.269 - 0.650i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (3.70 + 3.70i)T + 89iT^{2} \)
97 \( 1 + (-0.124 + 0.0834i)T + (37.1 - 89.6i)T^{2} \)
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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

−12.32709886663204182177499026176, −11.60350880682607988747461115129, −11.08885186164045336972047963723, −8.872229953613749137369078925618, −8.329206942818390823050810943352, −7.38481065826115120321225158891, −5.98550358789062583992901849353, −4.53634872378347019995555316706, −2.71215597996366692006690262167, −1.36419799462112842842294559959, 3.55538919960293098393084176184, 4.31114063915603394786522261328, 5.13637304981359138501217323363, 6.98421237837318705766309258167, 8.279428787136134427082370160725, 8.716222874325053086683404652541, 10.38426675144069157511737430823, 10.92026851308810798887501371730, 11.88009831550621855610369821987, 13.68639120777924461197103733046

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