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Properties

Label 2-170-85.12-c1-0-3
Degree $2$
Conductor $170$
Sign $0.908 - 0.417i$
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 + (0.0161 − 0.0811i)3-s + (−0.707 − 0.707i)4-s + (1.74 − 1.40i)5-s + (0.0688 + 0.0459i)6-s + (0.214 + 0.143i)7-s + (0.923 − 0.382i)8-s + (2.76 + 1.14i)9-s + (0.628 + 2.14i)10-s + (0.818 − 1.22i)11-s + (−0.0688 + 0.0459i)12-s − 0.161·13-s + (−0.214 + 0.143i)14-s + (−0.0857 − 0.164i)15-s + i·16-s + (0.561 + 4.08i)17-s + ⋯
L(s)  = 1  + (−0.270 + 0.653i)2-s + (0.00932 − 0.0468i)3-s + (−0.353 − 0.353i)4-s + (0.778 − 0.627i)5-s + (0.0280 + 0.0187i)6-s + (0.0810 + 0.0541i)7-s + (0.326 − 0.135i)8-s + (0.921 + 0.381i)9-s + (0.198 + 0.678i)10-s + (0.246 − 0.369i)11-s + (−0.0198 + 0.0132i)12-s − 0.0448·13-s + (−0.0573 + 0.0382i)14-s + (−0.0221 − 0.0423i)15-s + 0.250i·16-s + (0.136 + 0.990i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.908 - 0.417i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.908 - 0.417i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12062 + 0.245003i\)
\(L(\frac12)\) \(\approx\) \(1.12062 + 0.245003i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.74 + 1.40i)T \)
17 \( 1 + (-0.561 - 4.08i)T \)
good3 \( 1 + (-0.0161 + 0.0811i)T + (-2.77 - 1.14i)T^{2} \)
7 \( 1 + (-0.214 - 0.143i)T + (2.67 + 6.46i)T^{2} \)
11 \( 1 + (-0.818 + 1.22i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 + 0.161T + 13T^{2} \)
19 \( 1 + (-1.05 + 0.436i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (7.32 - 1.45i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-5.25 - 1.04i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 + (4.63 + 6.94i)T + (-11.8 + 28.6i)T^{2} \)
37 \( 1 + (5.47 + 1.08i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (5.60 - 1.11i)T + (37.8 - 15.6i)T^{2} \)
43 \( 1 + (0.798 + 1.92i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 9.14iT - 47T^{2} \)
53 \( 1 + (0.845 + 0.350i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (0.837 - 2.02i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-0.778 - 3.91i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (8.09 - 8.09i)T - 67iT^{2} \)
71 \( 1 + (8.03 - 5.36i)T + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (-2.58 + 1.72i)T + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (-0.595 - 0.398i)T + (30.2 + 72.9i)T^{2} \)
83 \( 1 + (-1.10 + 2.66i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-9.97 - 9.97i)T + 89iT^{2} \)
97 \( 1 + (-2.93 + 1.95i)T + (37.1 - 89.6i)T^{2} \)
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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

−13.07074010147980705101854288263, −12.02684892803346440361321765161, −10.47467872213272863286774353715, −9.772477222705554866635105926523, −8.688810327162613008772137536560, −7.75113837485195053568889654409, −6.42886052585308077190316302000, −5.46314338141773714327385846239, −4.18201922841535838569273260038, −1.68476013405346166619515695614, 1.80884686833437687216320391331, 3.38785981721603610918068674222, 4.88103990139905387698288296222, 6.48135563222043767607665983534, 7.46759946066762159213978914563, 8.969847732538001087462076733261, 9.954504457196401039672304715701, 10.41500539936873760616027512066, 11.75936607809039270092494291049, 12.52092616246607003133353620871

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