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Properties

Label 2-170-85.23-c1-0-6
Degree $2$
Conductor $170$
Sign $0.995 + 0.0959i$
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 + (1.41 + 0.942i)3-s + (0.707 − 0.707i)4-s + (2.21 − 0.293i)5-s + (1.66 + 0.330i)6-s + (−5.02 − 0.999i)7-s + (0.382 − 0.923i)8-s + (−0.0471 − 0.113i)9-s + (1.93 − 1.11i)10-s + (−0.999 + 5.02i)11-s + (1.66 − 0.330i)12-s − 3.13·13-s + (−5.02 + 0.999i)14-s + (3.40 + 1.67i)15-s − i·16-s + (3.59 − 2.02i)17-s + ⋯
L(s)  = 1  + (0.653 − 0.270i)2-s + (0.814 + 0.544i)3-s + (0.353 − 0.353i)4-s + (0.991 − 0.131i)5-s + (0.679 + 0.135i)6-s + (−1.89 − 0.377i)7-s + (0.135 − 0.326i)8-s + (−0.0157 − 0.0379i)9-s + (0.612 − 0.353i)10-s + (−0.301 + 1.51i)11-s + (0.480 − 0.0955i)12-s − 0.870·13-s + (−1.34 + 0.267i)14-s + (0.878 + 0.432i)15-s − 0.250i·16-s + (0.871 − 0.490i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.995 + 0.0959i$
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: \((2,\ 170,\ (\ :1/2),\ 0.995 + 0.0959i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92415 - 0.0924902i\)
\(L(\frac12)\) \(\approx\) \(1.92415 - 0.0924902i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 + 0.293i)T \)
17 \( 1 + (-3.59 + 2.02i)T \)
good3 \( 1 + (-1.41 - 0.942i)T + (1.14 + 2.77i)T^{2} \)
7 \( 1 + (5.02 + 0.999i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (0.999 - 5.02i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + 3.13T + 13T^{2} \)
19 \( 1 + (0.297 - 0.718i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-0.327 - 0.489i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (1.78 - 2.66i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (0.00940 + 0.0472i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (1.22 - 1.83i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-0.774 - 1.15i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (-2.55 - 1.05i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 1.90iT - 47T^{2} \)
53 \( 1 + (4.43 + 10.6i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-2.75 + 1.14i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (8.20 - 5.48i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (7.07 + 7.07i)T + 67iT^{2} \)
71 \( 1 + (-7.55 + 1.50i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-7.03 + 1.39i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (-9.46 - 1.88i)T + (72.9 + 30.2i)T^{2} \)
83 \( 1 + (15.6 - 6.48i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-5.07 + 5.07i)T - 89iT^{2} \)
97 \( 1 + (-1.77 + 0.353i)T + (89.6 - 37.1i)T^{2} \)
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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

−12.77078367722189338388850648944, −12.28145722083134664464610326661, −10.25649560525990035178812236979, −9.795571244903303358808161173731, −9.293250984762195057994091733489, −7.30844996646009883255410878434, −6.31374615140271147471914023361, −4.94407153254513799551302525858, −3.51527636752054173995854875614, −2.50721632736948288640481354960, 2.58071308184483012451263865926, 3.29422256452685590140974793364, 5.58404018319402190870245726019, 6.25456453516288206278263465694, 7.42039147577002797439112839846, 8.705782643675302638357055176983, 9.629711831856293397520726662827, 10.72223074103040993003461586046, 12.38467744826753826224615559348, 13.00796162040664853581818228950

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