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Properties

Label 2-170-85.37-c1-0-3
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: $\chi_{170} (37, \cdot )$
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

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

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