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Properties

Label 2-170-85.58-c1-0-6
Degree $2$
Conductor $170$
Sign $0.895 - 0.444i$
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 + (2.60 − 0.518i)3-s + (−0.707 + 0.707i)4-s + (−1.21 − 1.87i)5-s + (1.47 + 2.20i)6-s + (0.892 + 1.33i)7-s + (−0.923 − 0.382i)8-s + (3.74 − 1.55i)9-s + (1.26 − 1.84i)10-s + (−3.02 + 2.01i)11-s + (−1.47 + 2.20i)12-s + 3.47·13-s + (−0.892 + 1.33i)14-s + (−4.13 − 4.25i)15-s − i·16-s + (−4.09 − 0.483i)17-s + ⋯
L(s)  = 1  + (0.270 + 0.653i)2-s + (1.50 − 0.299i)3-s + (−0.353 + 0.353i)4-s + (−0.543 − 0.839i)5-s + (0.602 + 0.901i)6-s + (0.337 + 0.504i)7-s + (−0.326 − 0.135i)8-s + (1.24 − 0.517i)9-s + (0.401 − 0.582i)10-s + (−0.911 + 0.609i)11-s + (−0.425 + 0.637i)12-s + 0.963·13-s + (−0.238 + 0.356i)14-s + (−1.06 − 1.09i)15-s − 0.250i·16-s + (−0.993 − 0.117i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.895 - 0.444i$
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: \((2,\ 170,\ (\ :1/2),\ 0.895 - 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71271 + 0.401816i\)
\(L(\frac12)\) \(\approx\) \(1.71271 + 0.401816i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.21 + 1.87i)T \)
17 \( 1 + (4.09 + 0.483i)T \)
good3 \( 1 + (-2.60 + 0.518i)T + (2.77 - 1.14i)T^{2} \)
7 \( 1 + (-0.892 - 1.33i)T + (-2.67 + 6.46i)T^{2} \)
11 \( 1 + (3.02 - 2.01i)T + (4.20 - 10.1i)T^{2} \)
13 \( 1 - 3.47T + 13T^{2} \)
19 \( 1 + (5.49 + 2.27i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-0.403 + 2.02i)T + (-21.2 - 8.80i)T^{2} \)
29 \( 1 + (-0.814 - 4.09i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + (-4.50 - 3.01i)T + (11.8 + 28.6i)T^{2} \)
37 \( 1 + (1.08 + 5.44i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-2.21 + 11.1i)T + (-37.8 - 15.6i)T^{2} \)
43 \( 1 + (2.00 - 4.83i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 1.43iT - 47T^{2} \)
53 \( 1 + (7.58 - 3.14i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-1.18 - 2.86i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-14.2 - 2.83i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (-0.280 - 0.280i)T + 67iT^{2} \)
71 \( 1 + (-8.39 + 12.5i)T + (-27.1 - 65.5i)T^{2} \)
73 \( 1 + (-1.83 + 2.74i)T + (-27.9 - 67.4i)T^{2} \)
79 \( 1 + (-8.99 - 13.4i)T + (-30.2 + 72.9i)T^{2} \)
83 \( 1 + (-2.41 - 5.82i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-3.47 + 3.47i)T - 89iT^{2} \)
97 \( 1 + (1.11 - 1.67i)T + (-37.1 - 89.6i)T^{2} \)
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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

−13.02059805818187630965418502130, −12.41289520969359428139303588890, −10.85675592664091163477882067813, −9.136121387785960330262160044816, −8.589876904781762133731779818854, −7.959277011604236475721323578334, −6.79027341595292471972974160002, −5.06141233416559767406626067889, −3.94250695882098963973153562206, −2.32367505186307048840175017993, 2.36501188866064148999612911051, 3.47519724326484337740842761008, 4.35218805247773980055411334952, 6.40812100360286913469865436136, 8.026194156062519993864853523877, 8.415824269555628742601706678382, 9.853075524950565885954391227317, 10.73097907873597301122287991323, 11.43635205169038553455507481797, 13.12268862478569357086179369096

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