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Properties

Label 2-170-85.4-c1-0-2
Degree $2$
Conductor $170$
Sign $0.998 + 0.0457i$
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 + (1 − 2i)5-s + (−1 − i)6-s + (1 − i)7-s − 8-s − i·9-s + (−1 + 2i)10-s + (1 + i)11-s + (1 + i)12-s + 4i·13-s + (−1 + i)14-s + (3 − i)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 + (0.447 − 0.894i)5-s + (−0.408 − 0.408i)6-s + (0.377 − 0.377i)7-s − 0.353·8-s − 0.333i·9-s + (−0.316 + 0.632i)10-s + (0.301 + 0.301i)11-s + (0.288 + 0.288i)12-s + 1.10i·13-s + (−0.267 + 0.267i)14-s + (0.774 − 0.258i)15-s + 0.250·16-s + (0.242 − 0.970i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0457i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.998 + 0.0457i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.998 + 0.0457i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10361 - 0.0252315i\)
\(L(\frac12)\) \(\approx\) \(1.10361 - 0.0252315i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2i)T \)
17 \( 1 + (-1 + 4i)T \)
good3 \( 1 + (-1 - i)T + 3iT^{2} \)
7 \( 1 + (-1 + i)T - 7iT^{2} \)
11 \( 1 + (-1 - i)T + 11iT^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + (-1 + i)T - 23iT^{2} \)
29 \( 1 + (5 - 5i)T - 29iT^{2} \)
31 \( 1 + (3 - 3i)T - 31iT^{2} \)
37 \( 1 + (1 + i)T + 37iT^{2} \)
41 \( 1 + (7 + 7i)T + 41iT^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 + (-3 - 3i)T + 61iT^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 + (-5 + 5i)T - 71iT^{2} \)
73 \( 1 + (-1 - i)T + 73iT^{2} \)
79 \( 1 + (1 + i)T + 79iT^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (3 + 3i)T + 97iT^{2} \)
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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

−12.57770029277143203940798612033, −11.75835080359848482161474602238, −10.45411056985443318059315654876, −9.408660352072327745284470722577, −9.056645678738201117170623599350, −7.88266893666026732482711460922, −6.59311842322815813823235461349, −5.02158479735626705153511171456, −3.70619029077656432920417665381, −1.65870401694631459801251421648, 1.95183226077621285820039656705, 3.14425880851546873272857987591, 5.50429705291244738612543323185, 6.73006360911130935752534265473, 7.76002577991912302604844038393, 8.530638885889693371735128168558, 9.732840016200310498109780662634, 10.75585917222103581887500233158, 11.50550095622503468957749566825, 12.98297855813398328049697376026

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