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Properties

Label 2-170-85.23-c1-0-7
Degree $2$
Conductor $170$
Sign $0.371 + 0.928i$
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 + (−0.628 − 0.419i)3-s + (0.707 − 0.707i)4-s + (−0.448 − 2.19i)5-s + (−0.741 − 0.147i)6-s + (1.27 + 0.252i)7-s + (0.382 − 0.923i)8-s + (−0.929 − 2.24i)9-s + (−1.25 − 1.85i)10-s + (−0.546 + 2.74i)11-s + (−0.741 + 0.147i)12-s + 4.10·13-s + (1.27 − 0.252i)14-s + (−0.637 + 1.56i)15-s − i·16-s + (−2.50 + 3.27i)17-s + ⋯
L(s)  = 1  + (0.653 − 0.270i)2-s + (−0.362 − 0.242i)3-s + (0.353 − 0.353i)4-s + (−0.200 − 0.979i)5-s + (−0.302 − 0.0601i)6-s + (0.480 + 0.0955i)7-s + (0.135 − 0.326i)8-s + (−0.309 − 0.747i)9-s + (−0.396 − 0.585i)10-s + (−0.164 + 0.827i)11-s + (−0.213 + 0.0425i)12-s + 1.13·13-s + (0.339 − 0.0675i)14-s + (−0.164 + 0.404i)15-s − 0.250i·16-s + (−0.607 + 0.794i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.371 + 0.928i$
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: \((2,\ 170,\ (\ :1/2),\ 0.371 + 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20688 - 0.817371i\)
\(L(\frac12)\) \(\approx\) \(1.20688 - 0.817371i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.448 + 2.19i)T \)
17 \( 1 + (2.50 - 3.27i)T \)
good3 \( 1 + (0.628 + 0.419i)T + (1.14 + 2.77i)T^{2} \)
7 \( 1 + (-1.27 - 0.252i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (0.546 - 2.74i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 - 4.10T + 13T^{2} \)
19 \( 1 + (-1.17 + 2.83i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-4.19 - 6.28i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (3.51 - 5.26i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (0.355 + 1.78i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (-0.190 + 0.284i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (1.73 + 2.59i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (-7.35 - 3.04i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 7.90iT - 47T^{2} \)
53 \( 1 + (4.29 + 10.3i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-9.44 + 3.91i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-5.15 + 3.44i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (-0.944 - 0.944i)T + 67iT^{2} \)
71 \( 1 + (7.94 - 1.58i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-1.75 + 0.349i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (16.2 + 3.23i)T + (72.9 + 30.2i)T^{2} \)
83 \( 1 + (3.31 - 1.37i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-7.01 + 7.01i)T - 89iT^{2} \)
97 \( 1 + (-8.19 + 1.63i)T + (89.6 - 37.1i)T^{2} \)
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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

−12.85107893068590888256826711248, −11.57501218390390036800656054077, −11.12357333335440463102798323824, −9.499377677852589921558670688034, −8.588974916346286712970656254149, −7.19332486077230051162398519137, −5.90300388961628101908549023799, −4.91699885633862157933886874030, −3.67064363294107576392322354970, −1.47804921002914055851665703105, 2.71266277753175267687058162403, 4.11890665563034880248510727549, 5.46465408046191197073200984023, 6.43448643328679077228945849916, 7.64916019783899709844727415425, 8.638644073888261227468694828097, 10.45293915222911411447181484441, 11.10244275574049164425729307319, 11.68805199039771367335871640337, 13.26031807521108780566477506306

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