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Properties

Label 2-170-85.23-c1-0-0
Degree $2$
Conductor $170$
Sign $-0.967 - 0.252i$
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 + (−0.856 − 0.572i)3-s + (0.707 − 0.707i)4-s + (−1.44 + 1.70i)5-s + (1.01 + 0.201i)6-s + (−1.64 − 0.326i)7-s + (−0.382 + 0.923i)8-s + (−0.741 − 1.79i)9-s + (0.686 − 2.12i)10-s + (−1.12 + 5.67i)11-s + (−1.01 + 0.201i)12-s − 4.62·13-s + (1.64 − 0.326i)14-s + (2.21 − 0.630i)15-s − i·16-s + (−3.82 − 1.53i)17-s + ⋯
L(s)  = 1  + (−0.653 + 0.270i)2-s + (−0.494 − 0.330i)3-s + (0.353 − 0.353i)4-s + (−0.647 + 0.761i)5-s + (0.412 + 0.0820i)6-s + (−0.621 − 0.123i)7-s + (−0.135 + 0.326i)8-s + (−0.247 − 0.596i)9-s + (0.217 − 0.672i)10-s + (−0.340 + 1.71i)11-s + (−0.291 + 0.0580i)12-s − 1.28·13-s + (0.439 − 0.0873i)14-s + (0.572 − 0.162i)15-s − 0.250i·16-s + (−0.928 − 0.372i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.967 - 0.252i$
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: \((2,\ 170,\ (\ :1/2),\ -0.967 - 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0205915 + 0.160555i\)
\(L(\frac12)\) \(\approx\) \(0.0205915 + 0.160555i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.44 - 1.70i)T \)
17 \( 1 + (3.82 + 1.53i)T \)
good3 \( 1 + (0.856 + 0.572i)T + (1.14 + 2.77i)T^{2} \)
7 \( 1 + (1.64 + 0.326i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (1.12 - 5.67i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + 4.62T + 13T^{2} \)
19 \( 1 + (0.932 - 2.25i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.85 - 2.78i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (-3.70 + 5.55i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (0.914 + 4.59i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (0.910 - 1.36i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-6.23 - 9.32i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (-9.87 - 4.08i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 2.54iT - 47T^{2} \)
53 \( 1 + (1.76 + 4.25i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (9.68 - 4.00i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-3.66 + 2.45i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (1.91 + 1.91i)T + 67iT^{2} \)
71 \( 1 + (8.05 - 1.60i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (5.37 - 1.06i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (4.50 + 0.895i)T + (72.9 + 30.2i)T^{2} \)
83 \( 1 + (5.66 - 2.34i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (10.2 - 10.2i)T - 89iT^{2} \)
97 \( 1 + (-9.36 + 1.86i)T + (89.6 - 37.1i)T^{2} \)
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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

−12.92584387615437664934201109961, −12.09075542199444441268302817008, −11.28410211296869925069574938413, −10.04547977306516596174306731142, −9.444106388798492996976376282930, −7.72713411263266612105059700570, −7.08193755107254290241648784293, −6.22344228302856045175809733778, −4.49937868208170457939921059771, −2.59250277493089523461362591353, 0.18292550050051591166769545687, 2.87257901596897563749441479383, 4.56081256268796241561093276660, 5.77854499515005139982750834268, 7.26051821455215683355233649698, 8.501227807533991735676500673509, 9.093093880491342924386060212946, 10.59845600853776381546050162887, 11.05819332930766599582702117930, 12.21287834644243336186327874606

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