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Properties

Label 2-170-85.62-c1-0-0
Degree $2$
Conductor $170$
Sign $0.111 - 0.993i$
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  + (−0.382 − 0.923i)2-s + (−2.44 − 1.63i)3-s + (−0.707 + 0.707i)4-s + (0.0825 + 2.23i)5-s + (−0.574 + 2.88i)6-s + (−2.63 − 0.525i)7-s + (0.923 + 0.382i)8-s + (2.17 + 5.24i)9-s + (2.03 − 0.931i)10-s + (0.617 + 0.122i)11-s + (2.88 − 0.574i)12-s + 3.89i·13-s + (0.525 + 2.63i)14-s + (3.45 − 5.60i)15-s − i·16-s + (−4.11 − 0.231i)17-s + ⋯
L(s)  = 1  + (−0.270 − 0.653i)2-s + (−1.41 − 0.944i)3-s + (−0.353 + 0.353i)4-s + (0.0369 + 0.999i)5-s + (−0.234 + 1.17i)6-s + (−0.997 − 0.198i)7-s + (0.326 + 0.135i)8-s + (0.723 + 1.74i)9-s + (0.642 − 0.294i)10-s + (0.186 + 0.0370i)11-s + (0.833 − 0.165i)12-s + 1.08i·13-s + (0.140 + 0.705i)14-s + (0.891 − 1.44i)15-s − 0.250i·16-s + (−0.998 − 0.0561i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.111 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.111 - 0.993i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.111 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148548 + 0.132860i\)
\(L(\frac12)\) \(\approx\) \(0.148548 + 0.132860i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 + 0.923i)T \)
5 \( 1 + (-0.0825 - 2.23i)T \)
17 \( 1 + (4.11 + 0.231i)T \)
good3 \( 1 + (2.44 + 1.63i)T + (1.14 + 2.77i)T^{2} \)
7 \( 1 + (2.63 + 0.525i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (-0.617 - 0.122i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 - 3.89iT - 13T^{2} \)
19 \( 1 + (1.11 - 2.68i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.14 - 1.71i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (7.22 + 4.82i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 + (8.76 - 1.74i)T + (28.6 - 11.8i)T^{2} \)
37 \( 1 + (-1.80 + 2.69i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-3.52 + 2.35i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (3.68 - 8.90i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 1.02T + 47T^{2} \)
53 \( 1 + (-13.1 + 5.45i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-2.84 + 1.17i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-7.66 - 11.4i)T + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + (2.36 - 2.36i)T - 67iT^{2} \)
71 \( 1 + (2.62 + 13.1i)T + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (8.34 - 1.65i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (-0.558 + 2.80i)T + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (-0.661 - 1.59i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (5.74 - 5.74i)T - 89iT^{2} \)
97 \( 1 + (2.65 - 0.528i)T + (89.6 - 37.1i)T^{2} \)
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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

−12.90715485880152768514752815469, −11.71916749888197768081626632917, −11.25596197275256579153538370225, −10.33844759491269215554248633454, −9.264733910918566040298038844051, −7.40260686656681852802499892715, −6.73456143822013870463353606695, −5.81022546755736405525238164241, −3.93088915815029293362665708343, −2.03985834513860050293750284971, 0.23087169608878818068879331324, 3.99224797434368558992662782051, 5.21046675735084746039551772540, 5.85411036395823421799872657418, 7.00844327297189669061734747483, 8.768789982498846539336113216003, 9.480785012167779556427176569818, 10.45177211411674796546028616459, 11.38209195883442885073519566991, 12.67169380745876692622014146760

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