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Properties

Label 2-170-85.37-c1-0-1
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: $\chi_{170} (37, \cdot )$
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

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

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