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Properties

Label 2-170-85.23-c1-0-4
Degree $2$
Conductor $170$
Sign $0.231 + 0.972i$
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 + (−1.19 − 0.800i)3-s + (0.707 − 0.707i)4-s + (2.21 − 0.271i)5-s + (1.41 + 0.281i)6-s + (−3.32 − 0.660i)7-s + (−0.382 + 0.923i)8-s + (−0.352 − 0.851i)9-s + (−1.94 + 1.10i)10-s + (0.778 − 3.91i)11-s + (−1.41 + 0.281i)12-s + 4.10·13-s + (3.32 − 0.660i)14-s + (−2.87 − 1.45i)15-s − i·16-s + (−3.10 − 2.71i)17-s + ⋯
L(s)  = 1  + (−0.653 + 0.270i)2-s + (−0.692 − 0.462i)3-s + (0.353 − 0.353i)4-s + (0.992 − 0.121i)5-s + (0.577 + 0.114i)6-s + (−1.25 − 0.249i)7-s + (−0.135 + 0.326i)8-s + (−0.117 − 0.283i)9-s + (−0.615 + 0.347i)10-s + (0.234 − 1.18i)11-s + (−0.408 + 0.0811i)12-s + 1.13·13-s + (0.888 − 0.176i)14-s + (−0.742 − 0.375i)15-s − 0.250i·16-s + (−0.752 − 0.658i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.231 + 0.972i$
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: \((2,\ 170,\ (\ :1/2),\ 0.231 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.522145 - 0.412322i\)
\(L(\frac12)\) \(\approx\) \(0.522145 - 0.412322i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 + 0.271i)T \)
17 \( 1 + (3.10 + 2.71i)T \)
good3 \( 1 + (1.19 + 0.800i)T + (1.14 + 2.77i)T^{2} \)
7 \( 1 + (3.32 + 0.660i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (-0.778 + 3.91i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 - 4.10T + 13T^{2} \)
19 \( 1 + (-2.45 + 5.92i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (1.82 + 2.73i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (4.93 - 7.38i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (-2.05 - 10.3i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (-3.51 + 5.26i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-0.637 - 0.953i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (-4.26 - 1.76i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 0.236iT - 47T^{2} \)
53 \( 1 + (-0.352 - 0.849i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (3.06 - 1.27i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-0.930 + 0.621i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (-6.65 - 6.65i)T + 67iT^{2} \)
71 \( 1 + (-6.11 + 1.21i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-2.06 + 0.411i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (0.0457 + 0.00909i)T + (72.9 + 30.2i)T^{2} \)
83 \( 1 + (0.909 - 0.376i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-6.70 + 6.70i)T - 89iT^{2} \)
97 \( 1 + (-6.84 + 1.36i)T + (89.6 - 37.1i)T^{2} \)
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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

−12.67808971826206133331424570446, −11.30423480354918993112528838271, −10.64190790110620955395737922159, −9.245917360260280811806337296579, −8.875139811298855574179005412564, −6.87181620862460232278929507883, −6.41741464546155890713528883249, −5.47861812665783092593609425800, −3.12403868290873601890244147309, −0.850429107798015841078816600972, 2.12276825363726262721820585333, 3.95591653512676494649162953911, 5.82489778822189432396340469622, 6.36144698748741222430367728354, 7.950990121649176311123019887100, 9.524405242374416843860178805559, 9.794966221582489516283344041678, 10.78198224806506601868508394168, 11.79788758051944118105542046401, 12.92617443152193531826216840556

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