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Properties

Label 2-170-85.19-c1-0-6
Degree $2$
Conductor $170$
Sign $0.188 + 0.982i$
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  + (−0.707 + 0.707i)2-s + (−0.0978 − 0.236i)3-s − 1.00i·4-s + (−0.496 − 2.18i)5-s + (0.236 + 0.0978i)6-s + (−3.48 − 1.44i)7-s + (0.707 + 0.707i)8-s + (2.07 − 2.07i)9-s + (1.89 + 1.19i)10-s + (−3.09 − 1.28i)11-s + (−0.236 + 0.0978i)12-s − 0.0184·13-s + (3.48 − 1.44i)14-s + (−0.466 + 0.330i)15-s − 1.00·16-s + (2.88 − 2.95i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.0565 − 0.136i)3-s − 0.500i·4-s + (−0.222 − 0.975i)5-s + (0.0964 + 0.0399i)6-s + (−1.31 − 0.545i)7-s + (0.250 + 0.250i)8-s + (0.691 − 0.691i)9-s + (0.598 + 0.376i)10-s + (−0.933 − 0.386i)11-s + (−0.0682 + 0.0282i)12-s − 0.00510·13-s + (0.931 − 0.385i)14-s + (−0.120 + 0.0854i)15-s − 0.250·16-s + (0.698 − 0.715i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.188 + 0.982i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.188 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.509378 - 0.420958i\)
\(L(\frac12)\) \(\approx\) \(0.509378 - 0.420958i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.496 + 2.18i)T \)
17 \( 1 + (-2.88 + 2.95i)T \)
good3 \( 1 + (0.0978 + 0.236i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (3.48 + 1.44i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (3.09 + 1.28i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 0.0184T + 13T^{2} \)
19 \( 1 + (-4.04 - 4.04i)T + 19iT^{2} \)
23 \( 1 + (-2.00 + 4.83i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (0.238 + 0.575i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (0.958 - 0.397i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-1.49 - 3.60i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (2.68 - 6.47i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (-3.22 - 3.22i)T + 43iT^{2} \)
47 \( 1 + 9.78T + 47T^{2} \)
53 \( 1 + (-9.03 + 9.03i)T - 53iT^{2} \)
59 \( 1 + (-10.1 + 10.1i)T - 59iT^{2} \)
61 \( 1 + (-2.19 + 5.30i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 - 11.7iT - 67T^{2} \)
71 \( 1 + (3.61 - 1.49i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-9.33 + 3.86i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-4.92 - 2.03i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-0.848 + 0.848i)T - 83iT^{2} \)
89 \( 1 + 14.4iT - 89T^{2} \)
97 \( 1 + (-6.49 + 2.69i)T + (68.5 - 68.5i)T^{2} \)
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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

−12.79444569994202099921846965046, −11.65866896426289628338798188283, −10.00425620963294101398187194164, −9.711684870178412150327196490996, −8.382435754240953321363593494245, −7.38540040967021267293729712197, −6.32711482584953873915999897264, −5.06928520183765368173514347573, −3.47733393894512776377825740691, −0.74065206691150382216227364915, 2.50277126915193012884587003826, 3.61297389191219211287380005845, 5.48281096977328115065056497374, 6.98897545376901239759338276212, 7.71813180475205131654253461702, 9.276603663275739441366394957273, 10.10681266093417914175405704885, 10.74570362634238140892758008180, 11.91758025094538263231131224203, 12.91274149183896907711764558638

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