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Properties

Label 2-170-85.48-c1-0-2
Degree $2$
Conductor $170$
Sign $0.391 - 0.920i$
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 + (−0.793 + 0.530i)3-s + (−0.707 − 0.707i)4-s + (2.21 − 0.297i)5-s + (−0.186 − 0.935i)6-s + (3.70 − 0.736i)7-s + (0.923 − 0.382i)8-s + (−0.799 + 1.93i)9-s + (−0.573 + 2.16i)10-s + (−2.74 + 0.546i)11-s + (0.935 + 0.186i)12-s + 5.74i·13-s + (−0.736 + 3.70i)14-s + (−1.60 + 1.41i)15-s + i·16-s + (2.41 − 3.33i)17-s + ⋯
L(s)  = 1  + (−0.270 + 0.653i)2-s + (−0.458 + 0.306i)3-s + (−0.353 − 0.353i)4-s + (0.991 − 0.132i)5-s + (−0.0760 − 0.382i)6-s + (1.39 − 0.278i)7-s + (0.326 − 0.135i)8-s + (−0.266 + 0.643i)9-s + (−0.181 + 0.683i)10-s + (−0.828 + 0.164i)11-s + (0.270 + 0.0537i)12-s + 1.59i·13-s + (−0.196 + 0.989i)14-s + (−0.413 + 0.364i)15-s + 0.250i·16-s + (0.586 − 0.809i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.391 - 0.920i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.391 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.878162 + 0.581032i\)
\(L(\frac12)\) \(\approx\) \(0.878162 + 0.581032i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 + 0.297i)T \)
17 \( 1 + (-2.41 + 3.33i)T \)
good3 \( 1 + (0.793 - 0.530i)T + (1.14 - 2.77i)T^{2} \)
7 \( 1 + (-3.70 + 0.736i)T + (6.46 - 2.67i)T^{2} \)
11 \( 1 + (2.74 - 0.546i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 - 5.74iT - 13T^{2} \)
19 \( 1 + (0.119 + 0.288i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-2.65 + 3.97i)T + (-8.80 - 21.2i)T^{2} \)
29 \( 1 + (0.463 - 0.309i)T + (11.0 - 26.7i)T^{2} \)
31 \( 1 + (5.00 + 0.994i)T + (28.6 + 11.8i)T^{2} \)
37 \( 1 + (-5.72 - 8.56i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (7.70 + 5.15i)T + (15.6 + 37.8i)T^{2} \)
43 \( 1 + (3.44 + 8.32i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 5.61T + 47T^{2} \)
53 \( 1 + (6.08 + 2.52i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (3.14 + 1.30i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-5.67 + 8.48i)T + (-23.3 - 56.3i)T^{2} \)
67 \( 1 + (1.94 + 1.94i)T + 67iT^{2} \)
71 \( 1 + (-0.782 + 3.93i)T + (-65.5 - 27.1i)T^{2} \)
73 \( 1 + (0.841 + 0.167i)T + (67.4 + 27.9i)T^{2} \)
79 \( 1 + (-0.872 - 4.38i)T + (-72.9 + 30.2i)T^{2} \)
83 \( 1 + (-0.565 + 1.36i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (7.49 + 7.49i)T + 89iT^{2} \)
97 \( 1 + (-15.3 - 3.04i)T + (89.6 + 37.1i)T^{2} \)
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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

−13.29657726079293973951531495132, −11.71468469866732552362170020892, −10.84853075913945934366152002437, −9.957219564351964922233146291685, −8.830916370234255525539250592074, −7.80702319649162553870024911565, −6.61771464719828148346677228644, −5.16074543757057417715891942734, −4.82043676588183288110082190820, −1.95092672754879688935370169549, 1.44811216121901293624494710310, 3.06802159598754302183294361337, 5.18643887808681678536390031708, 5.84795869696290692773431793341, 7.63077768807331876162011494139, 8.524584897088513788062244519131, 9.777189852114208182528933160016, 10.75801615546827075088491812696, 11.39695180626757662360476805935, 12.64734392096434287808949979219

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