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Properties

Label 2-170-85.62-c1-0-4
Degree $2$
Conductor $170$
Sign $0.984 - 0.172i$
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 + (−1.16 − 0.776i)3-s + (−0.707 + 0.707i)4-s + (1.31 − 1.80i)5-s + (0.272 − 1.37i)6-s + (3.77 + 0.749i)7-s + (−0.923 − 0.382i)8-s + (−0.399 − 0.965i)9-s + (2.17 + 0.524i)10-s + (3.05 + 0.606i)11-s + (1.37 − 0.272i)12-s + 4.47i·13-s + (0.749 + 3.77i)14-s + (−2.93 + 1.07i)15-s − i·16-s + (−3.21 − 2.57i)17-s + ⋯
L(s)  = 1  + (0.270 + 0.653i)2-s + (−0.671 − 0.448i)3-s + (−0.353 + 0.353i)4-s + (0.588 − 0.808i)5-s + (0.111 − 0.559i)6-s + (1.42 + 0.283i)7-s + (−0.326 − 0.135i)8-s + (−0.133 − 0.321i)9-s + (0.687 + 0.165i)10-s + (0.919 + 0.182i)11-s + (0.395 − 0.0787i)12-s + 1.24i·13-s + (0.200 + 1.00i)14-s + (−0.757 + 0.278i)15-s − 0.250i·16-s + (−0.780 − 0.625i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.984 - 0.172i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.984 - 0.172i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24484 + 0.108253i\)
\(L(\frac12)\) \(\approx\) \(1.24484 + 0.108253i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.31 + 1.80i)T \)
17 \( 1 + (3.21 + 2.57i)T \)
good3 \( 1 + (1.16 + 0.776i)T + (1.14 + 2.77i)T^{2} \)
7 \( 1 + (-3.77 - 0.749i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (-3.05 - 0.606i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 - 4.47iT - 13T^{2} \)
19 \( 1 + (-1.25 + 3.02i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.36 - 2.05i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (5.13 + 3.42i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 + (6.04 - 1.20i)T + (28.6 - 11.8i)T^{2} \)
37 \( 1 + (4.21 - 6.31i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-1.67 + 1.12i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (2.41 - 5.82i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 9.31T + 47T^{2} \)
53 \( 1 + (11.5 - 4.76i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-4.66 + 1.93i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (1.54 + 2.31i)T + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + (6.06 - 6.06i)T - 67iT^{2} \)
71 \( 1 + (-0.137 - 0.691i)T + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (-1.32 + 0.264i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (2.06 - 10.3i)T + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (5.12 + 12.3i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-4.33 + 4.33i)T - 89iT^{2} \)
97 \( 1 + (-7.72 + 1.53i)T + (89.6 - 37.1i)T^{2} \)
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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

−12.82385712665609443874991239534, −11.69888599133680337699161212399, −11.40827613200954908984374391209, −9.276455225601302613081156052461, −8.861397961493905142803514296169, −7.36011873737450341946109705452, −6.36415121534161056412341759335, −5.27171219587902343019615333107, −4.40036726873754081214286906879, −1.61899804794579686914599747989, 1.90874059989789529884476927391, 3.75870799482099279170197064470, 5.12143485841678369751492796065, 5.94586182563819022704577205493, 7.54250215492414513264874988622, 8.869860591433497376091453437374, 10.28129154042894722854322768937, 10.89030659917989512114087161783, 11.33777290991688866196263166533, 12.61460458820042375069386845277

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