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Properties

Label 2-170-85.58-c1-0-7
Degree $2$
Conductor $170$
Sign $0.364 + 0.931i$
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 + (1.51 − 0.302i)3-s + (−0.707 + 0.707i)4-s + (1.28 − 1.82i)5-s + (−0.860 − 1.28i)6-s + (0.134 + 0.201i)7-s + (0.923 + 0.382i)8-s + (−0.554 + 0.229i)9-s + (−2.18 − 0.488i)10-s + (1.18 − 0.788i)11-s + (−0.860 + 1.28i)12-s − 0.250·13-s + (0.134 − 0.201i)14-s + (1.40 − 3.16i)15-s − i·16-s + (3.42 + 2.28i)17-s + ⋯
L(s)  = 1  + (−0.270 − 0.653i)2-s + (0.877 − 0.174i)3-s + (−0.353 + 0.353i)4-s + (0.575 − 0.817i)5-s + (−0.351 − 0.525i)6-s + (0.0507 + 0.0760i)7-s + (0.326 + 0.135i)8-s + (−0.184 + 0.0766i)9-s + (−0.690 − 0.154i)10-s + (0.355 − 0.237i)11-s + (−0.248 + 0.371i)12-s − 0.0695·13-s + (0.0359 − 0.0537i)14-s + (0.361 − 0.817i)15-s − 0.250i·16-s + (0.831 + 0.555i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.364 + 0.931i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.364 + 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09497 - 0.747420i\)
\(L(\frac12)\) \(\approx\) \(1.09497 - 0.747420i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.28 + 1.82i)T \)
17 \( 1 + (-3.42 - 2.28i)T \)
good3 \( 1 + (-1.51 + 0.302i)T + (2.77 - 1.14i)T^{2} \)
7 \( 1 + (-0.134 - 0.201i)T + (-2.67 + 6.46i)T^{2} \)
11 \( 1 + (-1.18 + 0.788i)T + (4.20 - 10.1i)T^{2} \)
13 \( 1 + 0.250T + 13T^{2} \)
19 \( 1 + (5.56 + 2.30i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.605 - 3.04i)T + (-21.2 - 8.80i)T^{2} \)
29 \( 1 + (-0.237 - 1.19i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + (-4.24 - 2.83i)T + (11.8 + 28.6i)T^{2} \)
37 \( 1 + (-1.26 - 6.37i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-0.171 + 0.860i)T + (-37.8 - 15.6i)T^{2} \)
43 \( 1 + (3.38 - 8.17i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 10.2iT - 47T^{2} \)
53 \( 1 + (-1.82 + 0.757i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-0.520 - 1.25i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (8.99 + 1.78i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (7.69 + 7.69i)T + 67iT^{2} \)
71 \( 1 + (0.548 - 0.821i)T + (-27.1 - 65.5i)T^{2} \)
73 \( 1 + (-8.70 + 13.0i)T + (-27.9 - 67.4i)T^{2} \)
79 \( 1 + (5.86 + 8.77i)T + (-30.2 + 72.9i)T^{2} \)
83 \( 1 + (0.0730 + 0.176i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-5.17 + 5.17i)T - 89iT^{2} \)
97 \( 1 + (-5.27 + 7.88i)T + (-37.1 - 89.6i)T^{2} \)
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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

−12.68196931947841855046568069373, −11.68063807508722311602367138403, −10.44814146188421508184941444983, −9.400248245756333546145295743672, −8.643629830700686884122577942833, −7.906026681669285894408581297119, −6.12363574022728471207732128565, −4.65506513271048880435986720292, −3.10775323473056638832299661521, −1.67680919169746037222942535746, 2.41664552500318096339608393288, 3.93527093295266900249859033679, 5.69744455689736570239415997167, 6.73644111104052192427614495124, 7.85653806257810120379683043918, 8.858244120992898709016990095086, 9.786120064820476393093634743894, 10.59828752732089656620492390962, 11.99593399750991823204840250929, 13.39119435003301204728049630583

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