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Properties

Label 2-170-85.23-c1-0-8
Degree $2$
Conductor $170$
Sign $-0.975 + 0.217i$
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  + (0.923 − 0.382i)2-s + (−2.47 − 1.65i)3-s + (0.707 − 0.707i)4-s + (−2.20 + 0.384i)5-s + (−2.92 − 0.581i)6-s + (−2.11 − 0.420i)7-s + (0.382 − 0.923i)8-s + (2.25 + 5.44i)9-s + (−1.88 + 1.19i)10-s + (0.581 − 2.92i)11-s + (−2.92 + 0.581i)12-s − 5.10·13-s + (−2.11 + 0.420i)14-s + (6.09 + 2.69i)15-s − i·16-s + (2.91 − 2.91i)17-s + ⋯
L(s)  = 1  + (0.653 − 0.270i)2-s + (−1.43 − 0.956i)3-s + (0.353 − 0.353i)4-s + (−0.985 + 0.171i)5-s + (−1.19 − 0.237i)6-s + (−0.799 − 0.159i)7-s + (0.135 − 0.326i)8-s + (0.751 + 1.81i)9-s + (−0.597 + 0.378i)10-s + (0.175 − 0.880i)11-s + (−0.844 + 0.167i)12-s − 1.41·13-s + (−0.565 + 0.112i)14-s + (1.57 + 0.696i)15-s − 0.250i·16-s + (0.706 − 0.708i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.217i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.975 + 0.217i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.975 + 0.217i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0620392 - 0.562363i\)
\(L(\frac12)\) \(\approx\) \(0.0620392 - 0.562363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.923 + 0.382i)T \)
5 \( 1 + (2.20 - 0.384i)T \)
17 \( 1 + (-2.91 + 2.91i)T \)
good3 \( 1 + (2.47 + 1.65i)T + (1.14 + 2.77i)T^{2} \)
7 \( 1 + (2.11 + 0.420i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (-0.581 + 2.92i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + 5.10T + 13T^{2} \)
19 \( 1 + (-2.47 + 5.97i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-0.847 - 1.26i)T + (-8.80 + 21.2i)T^{2} \)
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 \( 1 + (-4.04 + 6.05i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-2.01 - 3.01i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (1.73 + 0.718i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 4.65iT - 47T^{2} \)
53 \( 1 + (-1.11 - 2.70i)T + (-37.4 + 37.4i)T^{2} \)
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 \( 1 + (-9.62 + 1.91i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (15.1 + 3.01i)T + (72.9 + 30.2i)T^{2} \)
83 \( 1 + (-9.03 + 3.74i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (5.41 - 5.41i)T - 89iT^{2} \)
97 \( 1 + (2.63 - 0.523i)T + (89.6 - 37.1i)T^{2} \)
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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

−12.31618049279478229391854056014, −11.44808572958156322774426300913, −10.98083072801310372975310258120, −9.541696368482542510318414714596, −7.48118867140744629645029189061, −7.01764063369670203741397394779, −5.80580391069518191151642416746, −4.76720821123642133202963146183, −3.03837456130122967714704899530, −0.49931591216921874941434128351, 3.58498473555790398071548939839, 4.57505599325473036155495526909, 5.51799200782645642744459956407, 6.65768067680341386506479082738, 7.77261670264248086573259563967, 9.632741704460694444643860850075, 10.28270545463869730943625759372, 11.56102094535367433231057681409, 12.25721132553161984135254915313, 12.64283226572062987349293451085

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